The biggest problem that Percy faces is the fact that he is a new kid at Camp Half-Blood, the camp where half-bloods (kids with one mortal parent and one parent who was one of the twelve Olympian gods) go to be trained as demigods. Not only does he have to learn the ropes, he has to face the other campers. Chiron shows Percy the grounds and tells Percy about the cabins, one for each of the twelve gods, where children of those gods lived.
Other problems stem from him being the new kid. He is placed in the cabin of Hermes, which accepts all newcomers, because he is undetermined, which means that he did not know which god was his father. That immediately sets him apart from everyone else who knows their parent already.
He also runs into a bully, Clarisse, and her lackeys. They are daughters of Ares, the war god. Clarisse drags Percy into the girls bathroom, intending to give him a swirly, but the water from the toilet shoots out, arcs over him, and hits Clarisse in the face. Water from all the other toilets and the showers soak Clarisse and her friends and force them out of the bathroom.
So despite the fact that he is at a camp for demigods, he faces the same problem that any new kid at any school or place would face.