What are some conflicts in Percy Jackson and the Olympians: The Lightning Thief by Rick Riordan?

One example of conflict in the text is the antagonism between Percy and Nancy Bobofit at Yancy Academy.  They've had a history of poking at each other, literally and figuratively, but at the art museum, things come to a head.  Percy says, "I don't remember touching her, but the next thing I knew, Nancy was sitting on her butt in the fountain, screaming, 'Percy pushed me!'"  This episode helps to lead to his expulsion from school.

Then, of course, there's the conflict between Percy and Mrs. Dodds (one of the Furies), after the conflict with Nancy.  He says, "The look in her eyes was beyond mad.  It was evil," and she proceeded to attack him.  Were it not for the pen/sword that Mr. Brunner tossed to Percy, he would likely have followed Mrs. Dodds' orders to "'Die, honey!'"

Once he arrives at camp, it doesn't take long for Percy to conflict with Clarisse, daughter of Ares.  When they play capture the flag, she says, "'we don't care about the flag.  We care about a guy who made our cabin look stupid.'"  She tries to skewer him with her electrified spear, and one of her half-brothers slashes Percy's arm with his sword, leaving him for dead.  Luckily, the water he collapses in restores him. 

One of the most important conflicts of the novel is the one between Percy and Luke.  Percy has been told by the oracle, "You will be betrayed by one who calls you a friend," and this comes to pass when Luke betrays Percy.  Once Percy realizes this, he accuses Luke, who "stood calmly and brushed off his jeans" while the scorpion he called up creeps slowly up Percy's leg.  Luke finally tells him, "'Good-bye, Percy.  There is a new Golden Age coming.  You won't be part of it.'"  He thinks that Percy should be willing to abandon the Olympians, as he has, and work with Kronos to begin a new world order.

There are also conflicts between Percy and his step-father, the Minotaur, Medusa, Annabeth, Echidna, Ares, Hades, Zeus, even Percy himself (consider when he must decide on whom to use the pearls when he's in the Underworld).  The novel is riddled with conflicts involving poor Percy!