He might first find a place to hide, like an abandoned building. There, he starts to realize how difficult it is to be on his own. He has to find food and water by himself.
Perhaps he was influenced by some new friends he met who had a different lifestyle or view of the world. They might have convinced him that running away from home was an exciting adventure and a way to break free from the restrictions at home.
One possible reason could be that he feels misunderstood. In the camp, everyone has high expectations of him as a son of Poseidon. If he makes a small mistake, it might be blown out of proportion. For example, if he accidentally causes a small flood during a training session, instead of being understood as still learning control over his powers, he could be harshly criticized. Another reason might be that he's tired of the constant danger at the camp. Every day there seems to be a new threat looming, whether it's a monster attack or some sort of internal camp conflict. He might just want to get away and have some peace for a while.
It could be due to a conflict with a particular authority figure at the camp. If he has a major disagreement with a camp leader over how to handle a situation, like dealing with a new enemy. Percy, being headstrong, might decide to run away and handle it his own way instead of following the camp's rules.
One possibility in 'Percy Runs Away Fanfiction' is that Percy runs away because he is being hunted. There could be a powerful enemy after him, and he decides that running away is the best way to keep his loved ones out of danger. So he leaves his home and goes into hiding, maybe in a far - off land or a hidden magical place, and while he's away, he has to learn to survive on his own and also figure out how to defeat the enemy that's chasing him.
One possible reason could be the constant feeling of being an outcast in the village. Despite his efforts, many villagers still look at him with suspicion because of the Nine - Tails inside him. He might feel that running away would give him a chance to find a place where he is truly accepted.
He might go to his mother's place. It's a place where he has always felt safe and loved. His mother has always been his anchor, and even though it might be a risky move as monsters could track him there, it could be his first instinct. Another option could be to go to a big city like New York. There are so many people and places to hide. He could blend in with the crowds and use the chaos of the city to stay unnoticed. He could also find refuge in an abandoned building or a place that has some sort of magical protection that he discovers by accident.
Obviously, Percy would be the main character. But there could also be his friends who might be trying to find him after he runs away. For example, Annabeth could be a major character in the story as she would be worried about Percy and use her intelligence to track him down.
One possible reason could be that she feels misunderstood at home. In the wizarding world, Hermione is a very intelligent and studious witch. If her family doesn't understand or support her magical pursuits, she might feel the urge to run away. For example, if they constantly tell her to focus more on 'normal' things and less on her spells and potions, it could drive her away.
The villagers who once shunned Naruto might start to realize his importance. They would see how much they took his presence for granted. His absence could make them reflect on their past behavior towards him. Some might even join the search parties out of guilt or new - found respect.