Well, in many fanfictions, Ash's training before going to Johto can vary greatly depending on the author's imagination. Sometimes it's intense and focused, while other times it's more laid-back and includes unexpected adventures.
There could be a group of rivals rather than just one. A team of young Johto trainers who idolize the previous Johto League champions. They see Ash as a threat to their dreams of following in their idols' footsteps. They might gang up on Ash during battles or try to sabotage his journey in more subtle ways, like spreading false rumors about him in the Pokémon Centers. This would force Ash to not only focus on his Pokémon training but also on his reputation management.
He might encounter rare Pokémon that are unique to the Johto region, like Chikorita or Cyndaquil as his starter. He could also face off against the infamous Team Rocket, who always seem to be up to no good in the Pokémon world.
In some fanfictions, the team plasma experiments on Ash might be centered around his special powers. For example, if Ash has some latent psychic ability related to Pokémon communication that they want to unlock and control. They could be using all sorts of unethical methods to figure out how his mind works when it comes to Pokémon training and battles. And they might also be looking at his physical endurance and how he can withstand tough Pokémon battles without getting overly tired, perhaps to create super - trainers of their own.
When Ash catches Greninja, it might change the dynamics within his team. Greninja could be a bit of a loner at first, not really fitting in with the rest of the Pokémon. But over time, through shared battles and experiences, it could learn to work well with the others and even inspire them to improve.
Jessie and James are the obvious ones. They are the main members of Team Rocket. Jessie could be constantly trying to outdo Ash in Pokémon battles or contests of skill. James, on the other hand, might be more conflicted about having Ash around. He could secretly admire Ash's kindness towards Pokémon and start to have second thoughts about their Team Rocket activities.
Well, for one, Haunter has some really cool moves like Hypnosis and Dream Eater. When Ash catches it, his team's battle strategies would change. His other Pokémon would need to learn to work with Haunter. Also, Haunter's presence might intimidate some of the opposing Pokémon, giving Ash an edge in battles. It could also open up new exploration opportunities as Haunter might be able to access areas that other Pokémon on his team can't due to its Ghost - type nature.
One common theme is redemption. Many fanfictions focus on the idea of Team Plasma members realizing the wrong in their ways and seeking redemption, often with Ash's help.