This question is open to interpretation, as Bud's journey throughout Bud, Not Buddy has many events.
I would argue that these are the most important events:
1. Bud moves from the "home" to the Amos house
This move kicks off the entire story and gives the reader immense insight into who Bud is and the kind of life he's been living since his mother died.
2. Bud escapes the Amos house
Bud's escape from the Amos house is one of the most dramatic parts of the book. He fights a "vampire," reconnects with his friend Bugs, and even has his first kiss in his escape from the Amos house to Hooverville.
3. Bud meets Lefty Lewis
This is a real development in the story. Other than being dramatic and humorous (Bud also thinks Lefty is a zombie or vampire at first), meeting Lefty gives Bud his first taste of "family" and leads to Bud meeting Herman E. Calloway.
4. Bud meets Herman E. Calloway
This is the moment we've been waiting for since the start of the novel! Bud meets Herman E. Calloway and his band, although Calloway is not exactly what Bud was expecting. We also discover the mystery surrounding the bedroom Calloway is so touchy about...
5. Bud discovers the truth about Herman E. Calloway's identity
Bud finally discovers that Herman E. Calloway is not his father, but his grandfather. Bud receives his saxophone and truly finds his new family, identity, and sense of belonging.