Perhaps an easier question to answer would be, "What problems does Junior not go through?"
Throughout this funny, touching, and often heartbreaking book, Junior goes through a series of trials and tribulations, both on the reservation and in his new life at Reardan High School.
The first problem that Junior tells us about, apart from the fact that he was born with water on the brain, is the poverty faced by everybody on the reservation. He and his family are so poor that they cannot to take Junior's much-loved dog, Oscar, to the vet when he gets sick, so Junior's father's solution was to take Oscar out back and shoot him.
The other big problem that Oscar later faced on the reservation was being rejected by his best friend, Rowdy, after he decided to leave the reservation high school and transfer to Reardan, a "white" school.
Once he gets to Reardan, he faces problems such as a much longer trip from home to school, which he sometimes has to make on foot. Then, of course, there are the inevitable problems of trying to fit in at a new school, which were exacerbated by the fact that Junior was different from all the other students.
Perhaps the biggest trial of all is the death of Oscar's sister, which has a profound effect on every facet of Oscar's life.