The default for Pokémon-centric fics are for them to take place in a nondescript forest in a nondescript region.
Any character, be it an Author Avatar, an original character, or even an existing fictional character from any sort of media, waking up in either the standard Pokémon continuity to make the trainer's journey, or in the Pokémon Mystery Dungeon continuity transformed into a Pokémon themselves. Cases of the latter, and sometimes the former, often have some of the main character's family, friends, or even enemies transported into the world along with them. Sometimes the "trainer is late so they end up with a nonstandard starter Pokemon" plot is very, very common thanks to many a writer using Ash's own acquisition of Pikachu as a basis for their character's beginnings. Otherwise, the classic "one of the three traditional region starters" is used. Fics that try to avoid both have their most common methods be the character receiving their first partner by finding them injured and bringing them to health or have the Mon given to them by a family member.
In Pokémon fanworks of all kinds, the idea of Pokémon fusion has been a very popular idea for a long time, thanks it being a feature in other Mons series like Digimon, while the Pokémon series would only play with it twice (the Gen V trio of legendaries in Gen V has Kyurem able to fuse with Reshiram or Zekrom to become more powerful, and the Gen VII trio has Necrozma able to fuse with Solgaleo or Lunala).
Manaphy using Heart Swap on characters is another fairly common plot. This tends to happen between two human characters, especially in fics based on the anime that usually have Manaphy switching Ash and one of his companions, though that isn't to say that Pokémon-Pokémon swaps or even human-Pokémon swaps (in any of the continuities) aren't fairly easy to find as well.
A popular crossover idea is to take character(s) from Series X and feature out what Pokémon they would have if they were a trainer. Would they have only one or two, or a full team? Would their team reflect the powers and abilities the character has in their home series, or would it based on their personality, character theme, etc.? This is most often seen in the form of fanart, but fanfics that amount to "Series X but with Pokémon" or "Series X characters in the Pokéworld" are just as common.
Pokémon being portrayed as slaves and rebelling, a deconstruction/alternate interpretation of the premise of the series. Many of these fics may make use of the ending of the anime as originally planned by Takeshi Shudō, and thus find themselves based more on that continuity.
The protagonist being villainous or at minimum dark. The game version of Dawn is especially so associated with this plot due to her design making for an Eerie Pale-Skinned Brunette, combined with her source game's relationship with Darkrai, Giratina, and Team Galactic.
The protagonists and their rivals turning into a Dysfunction Junction after the events of the games. This is most common in games with darker plots like the Sinnoh, Unova, and Alola games, but it applies in general.
The following formula appears a lot for shipping fics: Misty, May, Dawn and Leaf get together with Ash, Drew, Paul, and Gary, respectively. Insert plot about generic high school drama/alter ego pop idols/secret agents/chosen heroes/and so on in between.
Probably the most popular set of plot points for fics related to the anime: the characters growing/being older and Ash is/or will become a Pokemon Master (whatever the author decides that means, usually being a Pokemon Master and being one of the Pokemon Champions is treated as basically one-in-the-same). This isn't specific to any single fic plot, but most of the fanfics tied to the anime, regardless of the broader plot, rely on the conceit that the characters can/do age. Particularly any shipping fics, because the idea of 10-year-olds hooking up (frequently followed by sex, shown or implied) is a bit much even for most hardcore shippers to accept. So it will quickly be established that time hasbeen passing as Ash engages in his adventures and the plot will go from there. In more recent non-AU fics, which usually take into account the most recent seasons, Ash and the other characters will usually be somewhere around 16-19, rather than canon-declared 10. Ash becoming a Pokemon Master is so common it's usually just a given with any fic with him as the main character. Frequently it will be a major part of the plot and his goal, other times he'll have already have achieved the title.
Tons of Ash/Misty fics are set up by Team Rocket kidnapping Misty and Ash saving her (or the other way around), leading them to discover their true feelings for each other.
Variations on plots where Ash is betrayed by his friends and loved ones and he goes off on his own with only his Pokémon (except in variants where even his Pokémon also betray him), becoming more powerful are unnervingly common. These fics tend to involve Ash vanishing for years before resurfacing at a tournament and defeating everyone with his new found strength. Also don't expect the betrayal part to make much sense or be in character. At all. With anyone. The "Betrayal Fic" is diffused enough it has finally started gathering parodies such as Reunions Can Be Good and Bad, and is generally considered horrible by a variety of writers up to the point of outright mockery.
Fics involving Ash getting serious with his training and goals. This usually involves Ash not behaving like Ash and basically being an author avatar for competitive battling. One specific variation of the "Ash Betrayed" plot is Ash being framed for some crime, being forced to go on the run and survive on his own, effectively forcing him to train harder just to survive and then meet up with the traitors at some League. Naturally these fics are Darker and Edgier than canon but also at least make an attempt to make sense beyond the characters betraying Ash because the plot requires them to.
Fics involving Ash's father and Ash's backstory. Some have his dad abandon his family when Ash was younger, others have him never even knowing his dad, and others have Ash's dad on a journey. There's also the sub-category of "Ash finds out that his dad is Giovanni", an idea that takes heavily from the stage musical Pokémon Live!, wherein Delia heavily implies this to be the case.
"*Enter Pokemon Here* Becomes A Human" fics are not uncommon, especially to work around Interspecies Romance.
Jessie reuniting with her Missing Mom Miyamoto is a popular concept in Team Rocket fics, as are Hurt/Comfort and angsty fics related to Jessie's trauma over Miyamoto's disappearance.
Ash becoming a Frontier Brain is a common plot that often comes with Ash/Anabel shipping.
There a lot of fanfics that include terms like "Smart!Ash" and "Powerful!Ash" in the summary (Naruto has a lot of similar fics). These are re-tellings of the anime featuring an Ash that isn't an Idiot Hero. They also very often either include the writer's favorite Ash ship (usually Ash/Misty or Ash/Serena) or are outright harem fics. An even larger number of these fics will have Ash basically being an author avatar and have little to any trace of his canon characterization.
Most fics about the Pokemon themselves are set in the games universe, however those that aren't tend to be about what the character's Pokemon do when they're not battling or training.
Fics that go into depth on Jessie's relationship with Cassidy are commonplace in TR-centric works.
In the Original Series days there were many fics about Oak and Delia have Unresolved Sexual Tension and having to come to terms with their feelings despite their large age gap (and usually baggage due to the loss of their previous lover). These haven't disappeared but are far less common than before.
A previous female partner meets up with Ash and it causes tension with Ash's new traveling partner. The most common set-up is Misty meeting Serena and either Serena realizing that Ash likes Misty or Serena otherwise being jealous of Misty's (romantic or platonic) relationship with Ash.
Prior to Ash's Pikachu being confirmed as male, many wrote it as female. In many of these works, Ash would be under the impression that Pikachu was male until given an Unsettling Gender Reveal. Oftentimes, Ash was the only one of his friends who didn't know this.
"What if [*insert companion's name here*] didn't leave Ash's group and remained with him for [*insert region name here*]?" Very common plot for shipping fics, for obvious reasons. Said companion will always be the one the author ships Ash with, and they will usually end up together not too far in. Usually these fics will just Handwave away whatever reason the anime gave for said character leaving Ash in the first place.
Patchwork Fics using the game's or Pokémon Adventure's characterizations for undeveloped characters in the anime's continuity.
It's not uncommon for fanfic writers to add to Ash's party several Pokémon he befriended but never officially caught, two of the most popular being Lucario from Lucario and the Mystery of Mew, and the Aura Sphere Riolu from Diamond & Pearl, or otherwise getting one of said evolutionary line. The latter became a bit of an Ascended Meme in the Pokémon Journeys anime, when Ash caught a Riolu for real.
Though it's not a plot but it hella appears more often in fanfics. The T.R.io (Jessie James and Meowth) often receive the short end of being used by authors/writers in Anime Fanfics since they tend to get Demoted to Extra. At best they appear lesser than usual while still relevant to the plots and at worst they end up as unimportant characters, cameos or Adapted Out entirely. The later will frequently have the author express dislike of them and their motto in the story in very blatant Author Tract moments.
Rewriting Ash as stronger and/or smarter, which in turn causes him to win controversial battles he lost.
There are quite a few in which Ash defeats Alain and takes the Kalos League Championship.
Ash learning to actively use his Aura abilities that came up in movie 8 alongside the Diamond and Pearl and Journeys series. As what that even entails isn't defined in canon this skillset basically becomes whatever the writer at the time wants it to do for the plot.
Pokémon Adventures: Red/Yellow fics where Red finds out that Yellow was the girl he rescued when he was ten.
Red accidentally kills Blue's Raticate and this either traumatizes Blue or makes him snap. The Trope Maker is likely Poor Rival.
Silver's mother being Ariana is a common fanon, so works about Ariana's relationship (or lack of one) wIth Silver and Giovanni aren't uncommon.
Black/White/Black 2/White 2
Hilda and N being reunited. Rarer is the alternative of using Hilbert instead of Hilda, though as mentioned above, this became a bit more common after Pokémon Generations.
Sun/Moon/Ultra Sun/Ultra Moon
Lillie returns from Kanto a few years in the future and pairs up with Selene/Eilio.
Red and Blue are on a honeymoon or date during the Alola games. Hijinks and romance occurs.
There are many fanworks on Lillie and Gladion's childhoods with Lusamine.
Post Ultra Sun and Ultra Moon, fics involving the Selene and Gladion pairing has stories of Gladion's travels in Kanto and Johto and how he got his Kanto starter, since it's Gladion who leaves for Kanto in Ultra Sun and Ultra Moon instead of Lillie.
A side-quest in the game that lets you obtain the apples needed to evolve an Applin to an Flapple or Appletun involves an urban legend that giving an Applin to someone they like would ensure a happy relationship between them. This has become a subject in various shipping fanworks where one gives the Applin to the other party.
Post-apocalyptic AUs where the Darkest Day happens but doesn't end.
Mixing elements of British mythology into Galar fanfics. Stories involving ghosts and fairies are especially common.