It is probably best to discuss two related but different resolutions for the book Bud, Not Buddy. The reason for discussing two different resolutions is because readers are walked through two different main conflicts over the course of the book.
One of those conflicts is about trying to find a family and a home. Bud is an orphan ever since his mother died, and the book begins with Bud being taken in by the Amos family. That ends disastrously, but Bud is blessed with families other that take him in quite lovingly. Lefty Lewis's family is a good example—however, Bud knows that he can't stay with them forever. He has to find a family of his own, and he does by the end of the book. Bud finds a home and a family with Jimmy, Doug, Eddie, Chug, Roy, and Miss Thomas. They are all members of the same band, and they warmly invite Bud in to be a part of their surrogate family.
The other resolution that is necessary to discuss is the resolution to Bud's search for his father. Bud's entire journey into Grand Rapids is spurred on by the fact that Bud believes that Calloway is his father. Bud ends up being wrong about Calloway. Calloway is not his father; however, Calloway is his grandfather. The story ends leaving readers with the impression that Calloway, the band members, and Bud are going to be a happy, functional family.