Percy would probably have to adapt to new powers and rules. Different worlds often have different magic systems. Maybe in this new world, water doesn't have the same properties as in his own world, and he has to learn to control a new form of elemental power.
He might encounter new and strange creatures. In many fanfictions, Percy could meet unique magical beings that don't exist in his own world. For example, he could come across sentient plants that can communicate or floating islands with their own mini - ecosystems.
He could form new alliances and friendships. There could be characters in the other world who share his sense of adventure or his values. They might team up with Percy to face common threats or to explore the mysteries of the new world.
Well, he might find that his demigod powers work differently. Maybe in this new world, the water doesn't respond to him in the same way as in his own world. He could also encounter new and strange creatures that he has never seen before, and have to figure out how to deal with them without the help of his usual friends from Camp Half - Blood.
In a 'Percy Jackson banished to another world' fanfiction, Percy would be in for a wild ride. He could find that the laws of physics and nature are completely different. For instance, gravity might work in a strange way or time could pass at a different rate. His skills as a fighter and strategist would be crucial. He may have to learn new languages to communicate with the inhabitants of this new world. And he'd probably have to face his own inner demons while being away from his loved ones and the familiar places of his home world.
In fanfiction, if Percy Jackson goes to highschool, he might face challenges like hiding his demigod powers from his classmates. He could also encounter new mythical creatures that are drawn to the school environment.
In a fanfiction where Percy Jackson goes to Azeroth, he could form alliances. Since he's used to being in a team in his own world, he'd likely team up with some of the heroes in Azeroth. For example, he could join forces with Thrall and help him in his quest for the Horde. His sword - fighting skills from Camp Half - Blood would come in handy in battles against the Burning Legion or other enemies.
If Percy Jackson goes to UA, there would be a huge culture clash at first. The UA students are used to a very structured hero training system, while Percy is more of a 'go with the flow' kind of demigod. But over time, he could teach them about improvisation in battle and they could teach him about teamwork in a more organized way. He might also introduce the concept of Greek mythology to the UA world, which could open up new ways of thinking about powers and abilities.
Well, he might get into all sorts of adventures. He could meet the dragons in Alagaesia and try to use his water powers to his advantage against them. Maybe he'd team up with the local heroes there and learn new combat styles.
Perhaps Percy would see a future where the monsters have evolved in strange ways. They could be more intelligent and organized, posing an even greater threat to the demigods. He might also find that some of his old friends have made great sacrifices and that he has a new role to play in this future world, like being a mentor to the new generation of demigods.
In this fanfiction, Percy might decide to break away from Camp Half - Blood. He could start exploring the mortal world on his own terms, using his demigod powers in ways that the camp rules don't allow. Maybe he teams up with some unexpected allies, like a group of outcast demigods or even some friendly monsters.
In this fanfiction, perhaps Percy will make new friends at Goode who end up getting involved in his demigod world. There could be a plot where the school is in danger, and Percy has to use his powers and knowledge of the mythological world to protect it. Maybe he'll also have some interesting interactions with his teachers who suspect there's something different about him.
Perhaps in this fanfiction, Percy goes to Australia to find a hidden artifact that is crucial for the safety of the demigod world. Along the way, he teams up with local demigods (if there are any in the fan - created story). He would have to adapt to the Australian culture and environment, which is very different from what he's used to in the US. He might also face some new types of monsters that are native to Australia.