Why do you think Spokane Indian Reservation and Reardan was chosen for this particular storyline?
Sherman Alexie has used the Spokane Indian Reservation as the setting for many of his stories, particularly in The Lone Ranger and Tonto Fistfight in Heaven and its movie counterpart Smoke Signals. He himself is a Spokane/Coeur d'Alene Indian who was raised on the reservation, and The Absolutely True Diary of a A Part-Time Indian is a semi-autobiographical work. It makes sense that Alexie would draw on his own real background and experience of growing up on the Spokane Reservation as the background for his story. He also left the reservation to go to school in Reardean, because he knew that getting a decent education on a poor reservation was not going to be possible for him. So that, too, is another authentic touch in his novel.
In addition to the ideas above, I would add that the purpose in these two settings is to highlight the cultural divide that exists between these two worlds and the narrator's attempt to navigate and make sense of each of those worlds and his movement between them. He is a "part-time Indian" for this very reason and he even acknowledges that he does not seem to fit within either world. The struggle here is not only in finding his own identify and sense of self-worth but in beginning to understand and appreciate the strengths and challenges of each of the environments in which he operates.