What challenges does Percy Jackson face in chapter 8 of The Lightning Thief?

Percy faces many challenges throughout chapter 8. The first challenge that we see is his struggle to find where he belongs in this new world. He has not yet been claimed by his father, so everyone is watching to see what happens. Percy becomes more bitter about this as time goes on:

"So okay, maybe gods had important things to do. But couldn't they call once in a while, or thunder, or something? Dionysus could make Diet Coke appear out of thin air. Why couldn't my dad, who-ever he was, make a phone appear?"

Often, demigods specialize in a skill that is related to their immortal parent. So he begins exploring different activities such as archery (associated with Apollo) and metal working (associated with Hephaestus). He fails at everything he tries.
Eventually, he finds some success in swordplay, which is its own special challenge. His teacher is Luke, who is the best sword fighter in 300 years. Percy has never lifted a sword before, and first has to spar with Luke. Yet, in the end, he is able to disarm his teacher once, to the surprise of everyone watching. 
The most prominent challenge that Percy faces is the Greek demigod version of "Capture the Flag". He is on the team that consists of the children of Athena, Apollo, and Hermes. Luke, who is also on this team, assigns him to border patrol, which should be a fairly simple task. However, within a few minutes of the game beginning, five members of the Ares cabin come to attack him as revenge for shaming them. All are experienced fighters, and Percy is forced to defend rather than attack until he is pushed back to stand in the creek behind him. He becomes reenergized as soon as he touches the water and is able to stand his ground until his team captures the flag. 
After an incident with a hellhound, Percy is able to overcome his first challenge: he is claimed by his father, Poseidon. With his place now known, the other campers all have a newfound respect for him (although children of Ares aren't particularly happy about it). However, this does not mean his journey has ended. It has only begun.