The first moment of awesome in the series was Red saving himself and Professor Oak from a crazed Machoke by reasoning on the fly that Bulbasaur, as a plant-based Pokemon, is empowered by sunlight and unleashing Solarbeam.
The Safari Zone chapters where in the end Red went from trying desperately to survive to capturing an army of Safari Pokemon.
Hell, Blue's battle against Sabrina. Looks like Sabrina's got her, but Blue retorts by sticking out her chest and taunts her saying that at least she's prettier. Of course, Sabrina's not going to take that from an 11 year old. so has Kadabra slash off Blue's top...to reveal that Blue's boobs were really Poke Balls, and Blue immediately releases her Pokemon for an unexpected attack.
You gotta admire three eleven-year-olds for pretty much taking down a criminal organization by themselves (not to mention taking down the entire building as well).
Also, the Red/Giovanni fight in its entirety. From Giovanni's strategy to Red's stealth release of Aerodactyl and his downright awesome comeback with Pikachu.
Red and Blaine's effort to capture Mewtwo. Bonus points to Aero for carrying the weight of Red's entire team at once. Note that this includes the half-ton Snorlax, not to mention Gyarados and Venusaur!
For the grand finale of the RGB arc, it comes down to Red's Venusaur versus his rival's Charizard. Everyone thinks it'll be an easy win for Charizard, but Red had purposely chosen his previous two Pokemon (Poliwrath and Pikachu) to set up for this match. The immense steam formed from the battle between Poliwrath and Ninetales and the electric attacks of Pikachu created thunderclouds overhead, so all that was left was to use Venusaur's Vine Whip to turn the flying Charizard...into a kite.
The fact that Red managed to do it with a combination of his three closest and longest-serving partners is a symbolically awesome moment (and awesome for the writer to make it not cheesy).
When Green and Bozz visit the Power Plant, Agatha appears and snatches Green up in a Dream Eater that traps him in his memories and slowly drains his life energy while he can't escape. Bozz is too scared to do anything, so Green sends a telepathic message to his Golduck, who relays it to Bozz via his Pokédex (a skill it had shown before) to basically tell Bozz to man up, all the while in extreme pain and barely able to stay awake. Even better, he gains that advice from the memories that Agatha trapped him in. Bozz then gets his own Moment Of Awesome as he does man up and use his Haunter to fight off Agatha, and then Green tops it all off by using his Scyther to slash through Agatha's Ghost Pokémon. Awesome.
Koga and Green are fighting Agatha and have to deal with her Arbok, who has special powers according to the pattern on it, which Agatha can change by touching it. Koga eventually counters this by having his Golbat use Leech Life on Koga himself, so Golbat can spit up the blood onto Arbok's hood, preventing Agatha from changing it, and leaving them helpless against Green's Golduck using a solid Confusion attack.
Blue's fight against Lorelei. After Lorelei uses voodoo to chain Blue to Sabrina, she spends part of the fight knocked out. After Sabrina herself is knocked out and thrown down a pit, Blue wakes up and uses Blastoise to escape the hole. Then she fights against Lorelei herself. Her first move is to have her Clefable use Minimize sneak up on Lorelei and steal the dolls. Unfortunately, Clefable drops the dolls and breaks them apart, severing Blue's arm. Except it then turns out that Blue's Ditto has been that arm the entire time. Turns out that Blue's been playing Lorelei (and Sabrina, who calls her out on this when she wakes up afterward) for fools the entire time. She was never chained to begin with, because she had made Ditto her fake arm before even entering the island cave because she felt the cold weather and predicted she'd face Lorelei in a situation like this. Damn, Blue's a magnificent bitch!
Red's return and rematch against Bruno. The whole battle was intense, but part that just won it for this troper was Red using the legendary super Evolution Stones to let Eevee switch between his three evolutions at will.
Yellow is fighting Lance, and just barely managing keep from falling into the lava that she's Surfing on, thanks to Lance's constant attacks. After a chapter's worth of dodging Lance's attacks, she decides to use what she learned from watching Mewtwo's fighting style and fight back... by Surfing in circles until she creates a lava tornado to suck Lance and his pokemon into the lava.
Immediately after, Lance and his pokemon survive a dip in the lava... using bubbles. Bubbles that take a Twineedle from Giovanni's Beedrill to cut through, but still. Bubbles.
And then of course there's the moment of awesome where all of Yellow's pokemon (sans Pika) evolve at the exact same time, in battle. Caterpie even evolves twice, all the way up to Butterfree.
Giovanni gets his own in his Big Damn Heroes moment when he shows up and proceeds to take down Lance, revealing that he too is a Viridian trainer, and the Beedrill mentioned above is a Viridian Pokemon.
Falkner deserves a mention here. Up until now only one Gym Leader had even come close to catching Suicine, and Falkner had been shown as a fairly decent new Gym Leader at best. Then not only does he nearly catch both Entei and Raikou, using a Pokemon that's a bad match up to both, but then when Suicine interrupts he almost manages to capture it too. Let me repeat that; he nearly managed to take down three Legendaries, at the same time, while at a disadvantage. Badass Normal indeed.
Gold falls into the ocean during the fight with Lugia, and the rope he was holding on to snaps. He's hundreds of feet away from any land or boats, the storm is making very large waves, and Lugia's Aeroblasting everything in sight. What does he do? Gold catches a wild Mantine, grabs his pool cue and about 20 Remoraids and turns the Mantine into a rocket propelled hang glider. He blasts out of the ocean yelling “I’M BACK!” and begins to dogfight with Lugia.
Green was trained by Chuck to hone his own abilities to be able to fight with his Pokemon. In his battle with Chuck he takes this to the Nth degree by focusing on Hitmontop's placement and using Mind Reader. I repeat. Green, through sheer force of will and concentration, uses a Pokemon move!
All of volume 13. The Gym Leader Exhibition Match in all its glory. And when it gets interrupted by Team Rocket, the Gym Leaders all go in charging in to stop them.
Villains Will and Karen torment Classy Cat-Burglar Blue with her phobia of birds by summoning Lugia and Ho-oh, the latter of whom caused Blue's phobia in the first place. After being shaken for a moment, Blue reveals she spent the past few months getting over her fear by tracking down and capturing some very particular bird Pokemon...Zapdos, Articuno, Moltres, I choose you!
Hell, literally everybody gets a Crowning Moment (also counts as heartwarming) during the finale of the the GSC arc. After being wounded by protecting Daisy from falling debris, Bill receives faxes from trainers all over region, them wishing that they could help in the battle at the Indigo Plateau, but can't send their Pokemon over since the Teleporter is still down. Bill gets a Heroic BSOD, crying out what good he is as a scientist since he can't fix the Teleporter, but as soon as Raikou's electricity powers up the machine, Bill wastes no time in sending out the message that it's working again. Every single minor character (and others obviously) introduced in the arc immediately rush over to send over their Pokemon, and a huge ass army of Pokemon, carrying their trainers' wills to stop the battle, manage to break Lugia and Ho-oh free from the Masked Man's control. The series never got more awesome than it was right here.
The final battle of the GSC arc also counts. Gold flies into the time stream riding on Raikou to stop Pryce. Then he tells Silver and Crystal to go back without him and he dives down towards the spot where Pryce is. Pryce asks how anyone could comprehend the pain he did when he held La Glace's egg in his hands after La Pris and La Prus died. Gold says he can, because he's the hatcher, then proceeds to unleash SUPER RISING THUNDER! (an attack powered by the moves of everyone else's Pokemon and channeled into his billiard cue like a lightning rod) and destroys the GS Ball that Celebi is stuck in.
Also, at the end of the arc, we learn that Crys has captured all 250 Pokemon native to Kanto and Johto. That's right, Crys managed to catch them all in her home continent.
Ruby's escape from Marge in the Rustburo Tunnel, which involves using his Pokéblock holder as an impromptu zip line.
Where Archie, acting as the head of the TV studio, walks out to his car, only to have the muffler start smoking. As he turns, he sees none other than Maxie, walking out of the flames that sprouted from it.
Words enough cannot describe how awesome it is when Ruby, who wears his hat for freakingly all over the time, even when sleeping, finally removes it!!!
Of course, the real awesome is when we flash back to how he got his scar that the hat is used to cover... as a child, he took on a Salamence. When all he had was Skitty, Poochyena, and Ralts. On top of that, he won.
The entire final battle of the RS chapter. Right up to Ruby summoning Celebi using a Hadouken pose. If you recall back to the beginning of the arc, you realize that Norman called out one of his Pokemon in the exact same manner!
Not as awesome, but there were some clever victories on Sapphire's hands prior to then. First, wasting Roxanne's Nosepass by luring it into facing north (after deducing which way north was by using the wind - smooth moves opening the door, Roxy) and exploiting Magnet Pull to accelerate Rono's Take Down. Second, demonstrating some minor skill at martial arts by having Toro counter Brawly's Hariyama with Double Kick - he only saw one at first, but discovered the second after moving Hariyama's hand while it was out.
And then there's Saph's battle with Ark and the simultaneous defense of Flannery. Ark used his Azumarill to flood the room and put an air bubble around his head - smart enough on its own, but he tried to make a bloody end for the both of them by calling out Sharpedo. She called out Rono to tank its Bite attack (and break its teeth), and then used his Evil Gloating time to cut open the window of their carriage to drain the chamber with the same teeth she broke off in the first place! Once Ark finally realized how boned he was, she had Rono Take Down him into the next car down! Yeah... that's what "owned" looks like.
Hell, the RS arc itself is acknowledged by fans worldwide (Japanese and Western fans alike) as the best arc up to this point in the entire series.
Blue's being assaulted by the unjustifiably angry citizens of the Sevii Islands while she's still weak and tired from the ordeal from her previous ordeal with Deoxys. However, the sailor from the Seagallop comes up with the ship, Kimberly throws down a rope, and Celio jumps at the mob all so that she can get away to continue fighting the real bad guys.
In terms of base level alone, Yellow's easily the weakest of all the Dex Holders (levels range from 20s to 40s). In the FRLG, when facing against Storc/Sird and Oca/Orm, the two of them pretty much laugh at her when checking up on her Pokemon's stats. However, she gets PISSED at the fact they're fighting at the Viridian Forest and her emotions make her Pokemon's levels RISE UP TO THE EIGTHIES. The two pretty much shit their pants and make a run for it, knowing that she'll OWN them if they try to fight back.
What makes this scene even more awesome is the fact that the person that calls the retreat and says point blank that Yellow would take them both down is freaking Sird, who thrashedLorelei earlier in the arc.
Technically, Yellow didn't actually get 'pissed', so to speak. Instead, her spirit rose. Which could be considered even more awesome, because her will to drive Team Rocket out of Viridian Forest was so huge that she quadrupled her Pokemons' levels, and yet she remained completely level-headed.
And Red's rematch with Giovanni. Mewtwo Vs. Deoxys. And it all starts with Giovanni on the lower ground, and he proceeds to demonstrate how utterly badass he is...
Giovanni: Welcome, Red and Mewtwo! How does it feel, looking down on me from where you're standing?
Red: What's that mean?!
Giovanni: You're on higher ground... Because the only one who has bested me in battle... Is none other than you, Red! I wish to make our positions clear... It is only fitting for me, as the challenger, to look up at you on lower ground. I stand here not as ex-gym leader of Viridian City, nor Team Rocket leader... But simply a trainer without any titles! Now... Come and face me, champion!
Pretty much everything Mewtwo does is a CMOA. One really awesome moment is when the three Dex Holders get grabbed by mechanical tentacles, the intent obviously to separate them. Mewtwo quickly brings out his spoon, extends it, and slices all three tentacles in one smooth movement.
Red stays behind on the flying fortress to disable the explosions and prevent it from blowing up over Vermillion City, after taking his coat off, and beating Deoxys and making it his ally and then making it transport everyone to safety, while staying behind himself. He truly proves his credentials as The Hero.
The climax of the Emerald arc...Just, goddamn the climax of the Emerald arc. The nine final-form starters of the first three generations unleash the elemental Hyper beams (Blast Burn, Hydro Cannon and Frenzy Plant) at a giant Kyogre made entirely of energy. And then Pika, Chuchu, and Gold's Pichu finish it off with a combined Volt Tackle.
All the dex holders fighting Guile Hideout's army of berserked Pokemon, especially the two-page spread of all ten of them together in the midst of battle.
Red's Poliwrath and Gold's Politoed become Bash Brothers and break Hideout's armor with a double Focus Punch.
Emerald's impassioned declaration that it's not battling Pokemon that he loves, but the Pokemon themselves. Doubles as a heartwarming moment, and it's what allows him to reach Jirachi to wish the petrified dex holders back to normal.
The unseen dex holder tournament to allow Emerald the chance to earn his final symbol... with the implication that he beat Red.
Dia calmly calling out Pearl saying that he has his own mind and won't be bossed around anymore, then both of them managing to thwart Cyrus together.
Diamond shows that he has balls of steel when he decides to stay back and face Cyrus to confirm whether or not the man is the leader of Team Galatic. Despite the fact Dia's shaking like crazy in fear (keep in mind that Cyrus nearly killed Pearl and Platinum in a previous event), he brings himself to sing the theme song of his favorite show to catch Cyrus's attention then ask him who he is and what his purpose is. And when Cyrus's response is to try to blow the kid up, Dia manages to steal his camera away from him.
Dia sneaking on-board Team Galactic's airship by himself. And then we get this. And later on he successfully infiltrates albeit temporarily TG's headquarters by disguising as a grunt and tries to find Uxie, Mesprit, and Azelf.
Dia really gets a lot of moments. When Dia calls out Cyrus for trying to destroy all of existence with a simple yet very heartfelt speech, Dialga and Palkia decide that they'd rather listen to him than Cyrus.
A little bit before that, Dia's own courage inspired Regigigas to fight Team Galactic. As Dia's teammate!
Regigigas proves to be legendary in every sense of the word when it puts Dialga and Palkia into a dual headlock.
The DP trio giving up their Pokedexes so they could free Uxie, Azelf, and Mesprit.
Cynthia vs. Cyrus! A battle everyone wanted to see.
When Sird makes an unexpected appearance on the Spear Pillar, she explains just what happened the last time we saw her: she had to flee all the way from Kanto to Sinnoh just to get Mewtwo off her trail. And this after she just revealed that she had a Darkrai of all Pokemon. Despite having the perfect counter to Mewtwo type-wise, she still fled from him. Mewtwo is just that badass.
Platinum's total breaking of Sapphire's badge-collection record. In the same amount of time Sapphire had collected three badges, Platinum had six.
Paka and Uji's defeat of Saturn. When it's time to get serious, goofballs they ain't.
Platinum's gym battle against Candice. With Rapidash close to fainting, Platinum only has Lopunny left, who quickly faints Abomasnow, but has a harder time against Candice's Froslass. Lopunny faints and Candice thinks she has won, only for Rapidash to come charging out of his pokéball and wiping the floor with Froslass. What happened? Knowing Froslass would faint her, Lopunny pulled a Heroic Sacrifice and fainted herself through the use of Healing Wish, which completely healed Rapidash! An Awesome Moment for both Platinum and Lopunny, as this was their first battle together.
Looker dodges a Gallade's attack from close-range. For a guy who's normally a comic relief, nonbattler... wow. No wonder he's in the international police.
Dia once again demonstrates his ability to empathize with Pokemon by managing to befriend Rotom and even convincing it to fight alongside them against Team Galactic.
Platinum's battles against the Sinnoh frontier brains. Now without the help of Diamond and Pearl beside her, she's shown to be able to hold her own in battle very well, what with having already obtained eight badges(and being the only one in the trio to do so). Her victories against frontier brains so far include Darach, Dahlia, Thornton, and Argenta. Impressive, indeed.
The field of meteorites at Veilstone? Turns out Cynthia's grandmother created it by practicing Draco Meteor in her youth.
Know how the battles between a few legendaries are pretty awesome? How about one between Shaymin, Giratina, Dialga, Palkia, Heatran, Regigigas, Darkrai, Cresselia, Uxie, Mesprit, and Azelf?
To top it all off, when Charon tries to escape the Distortion World using Giratina, he is immediately greeted by Cynthia and her Garchomp calling forth a storm of Draco Meteors.
Silver falling off a Rope Bridge...only to magnificently rise up standing on his Gyarados, demanding that Petrel hand over the plates.
When Archer tries to use Arceus' power to recreate the mythical dragon trio for his own usage, Silver provides distraction while Gold and Crystal team up to destroy the dragons before they're fully summoned. And then Arceus, free of Archer's control and understandably pissed, uses Judgment and blasts the four Rocket executives into oblivion.
Played a little differently in the official volume release. The dragons reform and the four Rocket executives were merely blasted back a small distance, but it's made even more awesome by seeing who comes in to the rescue: Giovanni, Lance, and Pryce, the first three big bads of the series, all ready to kick ass.
Pryce and Lance taking on their respective dragons. Pryce is immune to Dialga's time-based attacks due to having been trapped in the time void, while Lance has no problem with Palkia bending his Hyper Beams away because he can just bend them right back.
N basically beats the crap out of Black while giving him a Hannibal Lecture on why Black can't win. Black tells him to shut up and while he still ultimately loses, Black's Pokemon show N on how much they believe in their trainer.
Lenora quickly re-establishes herself as That One Boss as she thrashes Black right from the get-go.
A rare non-action example. Just watching White supervising the preparations for the first Pokemon Musical is a sight to behold. Who cares if she can't fight? Doesn't mean she doesn't have other strengths that are equally impressive.
Black finally has awakened Reshiram, but the Seven Sages are in the way. Unlike the games, however, six of the Gym Leaders were already captured by Team Plasma, Drayden and Iris are stuck protecting the civilians, and the Striaton Trio are too wounded from their battle with the Shadow Triad to show up. Who shows up to the rescue instead? Every named minor character Black has encountered during his journey, declaring that even if they aren't Gym Leaders or Elite 4 members, they still have dreams that they want to fight for and protect.
Rakutsu's first fight as a Pokedex holder, where he not only fights Corless' Genesect and actually wins, but finishes the fight by capturing it!
Series in general
Heck, nearly every single battle in the series has some sort of Crowning Moment, given how many of them revolve around complex and ingenious multi-pokemon strategies running off of Rule of Cool and the fact that the trainers themselves do way more than just bark commands.
You gotta give the manga-ka some credit for sticking with this series for as long as he has. Most Pokemon manga are just quick cash-ins or advertisements for the latest games, barely having enough content to discuss or speculate about. Hidenori Kusaka actually began when the franchise started and is still going strong with only Pocket Monsters being the other long-running Pokemon manga. Furthermore, due to the fact he doesn't work for Game Freak and plays the games to figure out how he wants to write his story, with all his Chekhov's Guns and Arc Welding, he is either the greatest master of planning things out or the greatest master of writing things out of his ass as he goes along. Either way, he's awesome.
The series is now doing well enough internationally that the creators got to go to France and Singapore for panels and autograph signings.