In One Piece, this trope applies to most characters. Devil Fruits don't get stronger, users become more creative at using them and develop secondary skills to make using them better. To cite one example, Crocodile of the Seven Warlords of the Sea is an Elemental Shapeshifter, having eaten a Devil Fruit that gives him the power to control, create, and turn into sand. He has mastered his Devil Fruit to the point that he can suck up the moisture from objects and people, leaving them as shriveled, withered husks.
In Bleach, the BadassBlood Knight Kenpachi is the epitome of Unskilled, but Strong, using absolute brute force to beat most opponents. But when he finally faces one whom brute force is useless against, he switches to his secret weapon: a basic kendo swing that Commander Yamamoto had forced him to learn... which is so powerful it causes his opponent toexplode. In chapter 520, Kyouraku's first order as the new captain-commander of the Gotei 13 is to give Kenpachi more formal training to prepare him for Vandereich's next move. Soul Society's ruling council actually doesn't want Kenpachi to get any stronger because they are worried he'll be unstoppable if he decides to rebel against Soul Society. Kyouraku convinces them that they need Kenpachi to become stronger, because if they lose the current war they'll be too dead worry about such things. And Kenpachi's teacher is Retsu Unohana the first Kenpachi.
In Dragon Ball, Goku lives for some amount of time with a colony of monkeys. The reason for this is because they use their tails for almost everything they do, and Goku's tail was his biggest weakness. By the time he returns to his friends, he is just as strong with his tail as with the rest of his body.
The officers of Section 2 in the Patlabor manga get trained in martial arts, to make them more versatile when piloting their giant humanoid mecha.
In Tiger & Bunny, Kotetsu spent much of his free time in high school learning how to fight without relying on his powers (which he limited to using only to help people in trouble), and managed to build a reputation as an undefeated street fighter without anyone knowing he was a NEXT. This comes in handy twenty years down the line when he starts losing his powers.
In Pokémon Special, Yellow quickly realizes that while her powers are useful, she needs actual battling skills in order to fight the Elite Four, so she asks Green to train her.
In Ratman, after the eponymous hero realizes that there are people out there that he can't defeat simply by being Unskilled, but Strong in his transformed state, he starts working out and taking boxing lessons in his civilian identity to improve his baseline strength and skill.
Kinnikuman usually got by on luck up until the American Tour Arc. In fact, his victory over Robin Mask in the Choujin Olympics (Widely considered by all to be the turning point of his career) was mostly due to Robin being too distracted going "My God, What Have I Done?" In Hawaii, he met Prince Kamehame, the former 99-time island champion. Kamehamne taught Kin his moves, including the iconic Kinniku Buster.
In one episode of Heartcatch Pretty Cure, Tsubomi and Erika confront a fellow classmate who knew martial arts. Turns out, the kid just copied stuff from movies and really didn't know how to do so. It's implied, though, they do start training earnestly when Itsuki joins the team.
In the Pokémon episode "Oshawott's Lost Scalchop!", Ash helps train his Oshawott to fight without his signature weapon, the scalchop, meaning Oshawott is no longer rendered helpless when he is disarmed.
Pikachu occasionally has to fight enemies who aren't affected by his electric attacks. He had to improve his speed to beat a Raichu, and he later learned Iron Tail.
In The Familiar of Zero, Agnes and her troops are ordered to train the students of Tristain Academy in swordsmanship and hand-to-hand combat. Montmorency said she didn't see the point... and then Agnes knocked her Magic Wand out of her hand and twisted her arm behind her back before she could react. The students then agree to the training.
In Attack on Titan, In a nutshell, this is what makes Titan Shifters so dangerous. Imagine fighting a fifteen-meter-tall Lightning Bruiser. Now imagine fighting one that's also a champion-level boxer.
Martian Manhunter has taught him techniques to help him resist mind control as that had been a problem for him.
He has learned Kryptonian martial arts as well. No, really. The son of Mongul also taught him a few things about fighting in an Enemy Mine situation.
In general, Superman's status as the superhero and his extensive history means he has the epitome of Taught by Experience going for him. In Justice League: Crisis on Two Earths, he remarks that he's more or less superior to Ultraman simply because he spends most of his time taking on the biggest threats in the 'verse, i.e. people who are stronger than him, so his skill as well as his strength is constantly tested.
He also tells Ultraman this in Trinity, stating that because he doesn't kill his enemies, he's constantly in action, unlike Ultraman who doesn't really need to work his skills that much.
This is one of the whole points to the X-Men: teaching mutants the ideal ways to use their individual powers to best help or defend mankind.
Wolverine is himself an example, having learned Japanese martial arts (including sword) well after he'd first exhibited his regeneration and superior senses. Depending on the continuity, it might even have been after he'd received the adamantium.
Elixir from New X-Men studied biology because his powers allowed him to alter organic tissue at will.
He also got all of Beast's medical knowledge telepathically implanted into his mind. Beast being the world's foremost expert in biochemistry.
Psylocke is a borderline example; while she has martial arts skills in addition to her Psychic Powers, and has the ability to use her powers and martial skills in unique ways, this was granted to her in an impossible Freaky Friday fashion, and not something she learned herself. Kwannon, the woman whose body she inhabits, is this trope played straight, as a low-level empath who trained to become an expert assassin.
During Planet Hulk, The Hulk becomes a gladiator in a planet with beings who approach him in sheer physical might, so he slowly but surely becomes a craftier fighter, using his strength in more ingenious ways instead of merely relying on it exclusively. The other Marvel "heroes" were in for a rude awakening during World War Hulk.
After discovering his powers, but before becoming Magneto, Max Eisenhardt studied numerous scientific fields, including mechanical and genetic engineering, as well as electromagnetic radiation. Because of this, Magneto is capable of instilling the mutant gene in ordinary people, creating Designer Babies from genetic tissue, and building technological marvels (such as an orbiting base on an Asteroid) that even ReedRichards admits are pretty impressive.
However, he is first and foremost a physicist, and naturally chose to specialize in electromagnetism and everything to do with it. Since his mutant power gives him control over the entire electro-magnetic spectrum, his scientific knowledge turns an already formidable mutant power into a full-fledged Green Lantern Ring- electromagnetism is one of the four fundamental forces of the universe, and knowing this and knowing just how few limits that really imposes on him makes Magneto one of the most powerful bastards on Marvel Earth and the sheer range of things he can do- up to and including opening interstellar wormholes and manipulating electrons to control even non-metals (eg. wood)- is absolutely mind-boggling to the layman who doesn't appreciate just how broken his power can be. He wouldn't be near that powerful if he didn't understand, in detail, the myriad ways his power works like that.
A good example of how important this is comes in the form of Magneto's youngest daughter Polaris...who has identical powers but none of his scientific expertise. This results in her being far less effective at using her control over magnetism.
More justified than usual in one issue of She-Hulk when Shulkie is in training to box the Champion of the Universe. She weightlifts as her human alter ego because she realises that increasing her strength as a human has a geometric effect when she Hulks Out.
Once, Beast Boy of the Teen Titans was seen watching Animal Planet to learn more animal forms he could turn into, and also what inherent powers he could gain from them.
Discussed in Ultimate Spider-Man, in which Peter states that he's been getting by on pure luck and reliance on his powers. Mary Jane then suggests he learn martial arts.
His classic universe counterpart finally did get some martial arts training from Captain America and Tony Stark tried to mentor him.
This also happened to Spider-Man in the main 616 universe. Having lost his Spider-Sense, he has begun to realize how much he depended upon it in battle, so he accepted an offer to train under Shang-Chi, one of the Marvel Universe's greatest martial artists. Shang-Chi is now helping Spider-Man develop a unique fighting style called "The Way of the Spider", that incorporates Spider-Man's strength and agility. And then in Spider Island, he regains his Spider-Sense and combines it with the Way of the Spider.
Like Superman above, Spider-Man's vast history and being completely Taught by Experience means he did develop extensive street-fighting experience.
Iron Man: Tony Stark, unable to access his armor for the time being, goes to Steve Rogers to learn how to fight during the Demon In A Bottle story arc, under the pretense of being more vulnerable to his enemies now that his "bodyguard" Iron Man won't be able to protect him. It comes in handy when Stark is captured by Justin Hammer and his goons.
Justin Hammer: Have you any questions?
Tony Stark: As a matter of fact, I do. I'd like to know if this guard here knows what a clavicle is.
Guard: Huh? Well, uh... no.
Tony Stark: Surprise! It's what I just broke!
One issue of the Green Lantern Corps had Kyle Rayner encounter a situation in which he could not challenge Sinestro without breaking a truce that kept a fragile peace. Kyle proposed that the two fight as normals—without anyGreen Lantern Rings. Sinestro accepts, brags that he had been trained by some of the greatest martial artists in the known universe, and asks Kyle who trained him. Kyle's reply? Batman.
In a two-page splash, no less.
Gen 13. Caitlin Fairchild had Super Strength and superhuman agility and speed, but in one issue Sarah Rainmaker started to teach her how to fight.
In Preacher, not learning how to fight has major consequences for Cassidy, who - despite being a vampire, stronger and faster than a human could ever be - finds that Jesse, who suffered through a protracted bout of Training from Hell, can hold his own when they go toe-to-toe. Jesse himself has a lack of training in one area - he doesn't speak French (or any language other than English), which turns out to be a bigger problem than you might imagine.
In Irredeemable, not ever learning how to fight and relying only on his strength alone come back to bite Plutonian hard, when somebody equally strong, who knows some moves, showed up.
In the New 52, the new Ray studies up on light after getting his powers in order to make himself effective with them.
Subtly mentioned in various X-Men titles, like the X-Men Evolution note below, Scott has learned various skills to compensate and enhance his power. He specifically trains to fight so that he doesn't lose his visor, has learned to fight blind and remember where his visor fell, and is a master of applied geometry. The latter has resulted in Wolverine owing Scott a truckload of beer over billiards.
The original Firestorm, whose powers involve molecular transformation, has studied chemistry to make them more effective. Particularly notable in that he's a jock who does not find study easy.
Taskmaster. His power? Being able to copy any move that he can physically perform as well as the guy he watched. The skill he later picked up? Teaching what he knew.
In depictions where Doctor Strange didn't learn the martial arts at the same time he learned magic, he picked them up later (often with the help of Wong) so that he could defend himself even without magic.
In Mystic, the Seven Spirits get fed up with Giselle's bratty party-girl attitude and quit helping her with magic, forcing her to learn about their schools of magic in order to actually earn their respect, and learn to use magic without them. In doing so, Giselle improves her own skills, builds new rapport with the Spirits, and takes a few levels in maturity, all of which make her more effective down the road.
In Game Theory, Nanoha learns staff fighting so that she can use her Device as a melee weapon in magical combat.
The central premise of Duel Nature is giving these to Twilight Sparkle and watching what happens.
Quirrell: Most wizards do not bother much with what a Muggle would term martial arts. Is not a wand stronger than a fist? This attitude is stupid.
In Iron Man 2, Tony is shown practicing "mixed martial-arts" (actually DirtyBoxing) against Happy Hogan. Later, when he and Rhodey get into a fight, the lessons actually come in handy because Tony is more used to fighting in Powered Armor and is actually able to hold his own against the military-trained Colonel Rhodes despite being drunk.
Bruce Banner in The Incredible Hulk is using Brazilian Jiu-Jitsu lessons to help him control his rage issues, possibly even to induce and control the Hulk, as suggested by his Love Interest, Betty.
In the Star WarsExpanded Universe, Jaina Solo asked Boba Fett to train her in Jedi dispatching technique and Mandalorian attitude (including being a Combat Pragmatist) to prepare for her final duel with Darth Caedus.
The Dresden Files: One of Harry Dresden´s manylevels in badass involves him learning basic martial arts and quarterstaff fighting from Murphy, as well as running periodically so he has options other than blasting things in the face. This pays off whenever he's fighting someone who his magic is ineffective against, or humans, who he can't use magic against without risking breaking the first law of magic.
Harry also notes that teaching his apprentice gave him a much better appreciation for the basics of magic he himself learned and took for granted, as well as gaining proficiency in areas of magic he's weak in (like illusion magic, which his apprentice excels in).
The investigative skills he acquired as apprentice to a detective frequently prove more useful than the ones he'd picked up as a student wizard.
Harry takes up lock-picking as physical therapy for his burnt hand, then uses it to conserve his magical energy and cause less property damage while breaking into places covertly.
Pewter-burners in the Mistborn books burn their metal to gain inhuman levels of strength, speed and stamina. However, this only lasts as long as the metal supply, and the power works independently of the burner's body - as in, they don't suddenly sprout muscles. Pewter-burners like Hammond instruct the main character, Vin, of the benefits of honing one's body without pewter to increase one's strength while using it, developing fighting skill alongside these abilities, and using pewter tactically for powerful strikes instead of turning it on all the time in a fight and flailing your limbs.
In Wearing the Cape, Hope Corrigan gains the Atlas-type power set, enabling her to outfly jets, bench-press buses, and take direct hits from military ordinance. So the first thing she does is go into intensive, fight-club style training so she has a chance against all the other Atlas-types out there.
In Liar, Micah was born being able to run abnormally fast. Then she gets taught proper running technique and is able to run that much faster and more efficiently. She's actually almost disgusted when she sees someone like her without technique and hopes she didn't look that bad before she learned.
The third book of the Septimus Heap series is partially about Septimus learning to become a Physician while trapped in a Time past. This becomes useful upon his return to his Time to control an epidemic.
Harry Potter: Harry, Ron and the Weasley twins can pick locks the Muggle way.
Asha'man in The Wheel of Time learn swordfighting alongside channeling. Rand (himself one of the most powerful channelers alive and eventually a blademaster) insists it be included in the curriculum specifically because magic isn't always an available option.
Sisters of the red ajah train in unarmed combat for the same reason.
More literally, Loup of Santa Olivia dedicates herself to her boxing training in order to defeat another person with the same superstrength as her, but who hasn't trained because he's coasting by on his (un)natural abilities.
Live Action TV
In an episode of No Ordinary Family, Jim Powell (who has Super Strength) confronts another guy who has also gained that superpower, and gets his ass handed to him because the other guy actually knows something about fighting. So that it doesn't happen again, sidekick George teaches Jim a few of the basics of boxing, and Jim wins the second round.
On Heroes, Claire demands that her dad, a Badass Normal, teach her how to fight and defend herself. So he hands her a 2x4 and teaches her a stance.
In one episode of Lois and Clark, Superman took a crash course in kung fu to face off against a martial artist who had stolen a mystical artifact that multiplied his strength.
The martial artist yielded after realizing he couldn't beat Supes (who now has strength and technique), although only after his teacher told him to do the honorable thing.
In Super Sentai, several series have an episode that focuses on the featured Ranger trying to learn the mundane skill the Monster of the Week uses a weaponized version of - more than once, it's actually boxing.
On that subject, in Mighty Morphin' Power Rangers, Billy ends up taking Martial Arts lessons from Jason once he becomes the Blue Ranger. It really doesn't start to show until the second season.
After the events of the Season Eight finale of Smallville, Kal-El declares his Clark Kent persona dead and leaves for the Fortress of Solitude to begin training under Jor-El in earnest. It pays off the next season as he's Brought Down to Normal more than once and has to rely on skill rather than his powers.
Gen Ootori in Ultraman Leo learns various techniques (mostly kung-fu-based) to better fight giant aliens and kaijuu.
In the official Champions setting, the French supervillain Venin Vert studied chemistry so as to be able to use her powers (she can produce poisons from her hands) more efficiently.
There are several RPG systems, for example the New World of Darkness, where the die roll for activating supernatural powers is tied to mundane skills. So improving the mundane skill (e.g. Persuasion) also makes the supernatural power (e.g. Summoning) more effective.
In Dungeons & Dragons and Pathfinder, monsters generally have superpowers as part of their innate nature. However, intelligent monsters might actually begin learning to be more effective by studying the same classes as a Player Character might. Depending on the edition of the game, this can result in them suddenly becoming far, far more threatening. In one notorious instance, a succubus who went from Chaotic Evil to Lawful Good took up being a Paladin, using her already superior physical abilities, natural resistances and spells, lengthy list of skills, and ungodly Charisma to become a very deadly enemy indeed. Her intrinsic fiendish nature (evil subtype) also protects her from attacks that target good-aligned characters, the bane of many paladins.
In Mabinogi, player characters gain stat points from training skills, including noncombat skills. This leads to players training, for example, the Weaving skill to use the Dexterity it rewards to help in combat. This also works in reverse, as some noncombat skills benefit from higher stats that can be gained by training skills, including combat skills.
The post-prologue beginning of the first Golden Sun game suggests briefly that Isaac, Garet, and Jenna were learning about Alchemy from Kraden to understand their powers better. Isaac's mother is baffled, since Alchemy itself is a dead power and supposed to remain that way, in theory...
This could apply to Pokémon as well. For example, you can teach one of your Pokémon how to swim and thus teach it a useful water attack.
Played straight by Effort Values, tiny stat boosts gained by fighting other Pokémon. Simply fighting random Pokémon and gaining EXP would yield a powerful monster, but with specialized EV Training focusing on specific stats, a Pokémon can yield monstrous gains in a stat.
Also to the players that use them. A person can just go through the game, train his Pokémon to level 100, and beat his friends. Unless one of his friends knows how to play competitively, and battles with EV trained, IV bred Pokémon with competitive move sets.
In Deus Ex: Human Revolution, following his conversion into a Hollywood Cyborg, Adam Jensen undergoes therapy to help him cope with his new cybernetic body. Part of the therapy includes Jensen teaching himself how to build and repair clocks, in order to relearn precision control of his new prosthetic hands. This is based on actual physical therapy techniques.
He still occasionally slips up, as shown in the trailer, where he's poured himself a shot of whiskey and the glass is shown to be slowly covering in cracks from his tight grip.
Pool Powers, most literally the Fighting Pool in City of Heroes, are one way characters can get some extra skills. Temporary powers are another.
In Academagia The Making Of Mages, the character is a student at a Wizarding School, but they also train in mundane skills to enhance and supplement their magic. For example, a student at Avila, the college of Astrology, is also required to study Geometry, because mundane Astronomy classes are part of the Geometry curriculum, and knowledge of mundane astronomy can assist in learning about magical astrology.
In Knights of the Old Republic II both Kreia and the Mandalorians note that most Jedi aren't particularly skilled at anything, and their only advantage is their connection to the Force that gives them their abilities. They train the Exile to not rely on the Force to provide her abilities, but to hone a more mundane skill and use the Force to augment it.
The Handmaiden also teaches the Exile hand-to-hand combat. In return, you can train her as a Jedi.
Spinnerette gets trained in kickboxing by Mecha Maid. Despite Spinnerette's Super Strength, she got beat up by a supervillain with no combat abilities (at the time)... and by an ordinary mugger.
In Gunnerkrigg Court, the main character, Annie, is a powerful magic user. However, because she was raised in a hospital, she never learned much about the outside world. She ends up spending one summer break with a family of forest pixies, to learn about family values, nature, and self-reliance.
In the Global Guardians PBEM Universe, the super-heroic Geomancer discovered his power to manipulate rock and stone while he was still in high school. He later went to college and earned a doctorate in geology (with a minor in geochemistry) specifically to improve the usefulness of his powers.
Arguably the entire purpose of Whateley Academy is to churn these types of characters out. Specifically, there are seven different martial artists, all of different schools, who teach classes. Students have to take either beginning aikido or 'Survival' for a Physical Education credit. Special classes teach everything from 'team tactics' (think Spec Ops applied to superheroes) to designing a costume to understanding how not only your power works but also how your opponent's power works.
Quite common in Worm — this is one of the standard benefits of The Wards, the US's superhero organization for minors.
A villainous example occurs in Justice League. Shade, who has a nightstick that creates shadows and manipulates them, is essentially rendered powerless without it. Batman beats him this way the first time. When it comes time for a rematch and Batman uses the same tactic, Shade sucker punches him, having taken lessons in preparation for this sort of thing. He still got his ass kicked, of course, but at least he learned.
In Young Justice, after being humiliated by Black Canary with a judo toss, Superboy begrudgingly takes private fighting lessons. The results start showing less than two episodes after.
The Disney version of Hercules had Herc go to Phil for training. He already had Super Strength, but Phil taught him how to use that strength in a real fight, along with other skills like archery and swordsmanship.
Used in an episode of X-Men: Evolution. Most of the younger mutants who are used to training in the danger room with Wolverine are disappointed when Scott and Jean are the only teachers left at the institute and would rather teach them geometry and physics. Of course, they change their minds when Scott shows them he can ricochet his laser eyes around a room to hit a moving target.
Discussed but not used in Iron Man: Armored Adventures. When Tony has trouble with an agile villain, Rhodey suggests he might need to learn kung fu. Tony complains that he already has a highly-expensive suit of Powered Armor.
Which leads to Tony Dramatically Missing The Point, as Steve's entire intention was to show that knowing how to personally defend yourself will never, no matter how far technology advances, become obsolete. Tony doesn't see what Cap means.
Apparently the lesson was not lost on him, as we see in the next season that the training is continuing:
happy Tony Stark: "And hey, I only blacked out the one time."
Skrull Steve Rogers: "If that's the way you choose to remember it..."
In Teen Titans everyone from the planet Tamaran has superhuman strength; however, Blackfire complements that by also knowing some Tamaranean martial arts. Also, Cyborg's mechanical body wouldn't be nearly as awesome if he weren't also a mechanical genius able to repair and upgrade it.