Prisoner of Azkaban is my favorite. It's everyone's favorite, but only because it's so great. A lot of cool elements were introduced, like the Knight Bus, Hogsmeade, and the Marauder's Map. I find the part where Fred and George introduce the map to Harry to be intriguing, because it's cool hearing them describe all the different Hogwarts secret passages.
Also, I love Crookshanks and the small side story of him helping Sirius, which the latter reveals in the Shrieking Shack. Plus, when Harry tries to kill Sirius, Crookshanks out-dogs Sirius by loyally jumping on his chest to protect him. That cat was like, "Ya wanna get to him, ya gotta go through me!"
And Ron standing up on his BROKEN LEG to protect Harry from Sirius is awesome. Saw a Tumblr post once that analyzed a quote in that scene: "Something flickered in Black's shadowed eyes." The posting fan asserted that flicker was Sirius realizing Harry had found friends just as loyal and willing to die for him as he did in James and Remus.
In that same set of chapters, Lupin "embraced Black like a brother" and I love, as a reader, going, "WHAT?!" and then I found out Scabbers is really Wormtail and I go, "WHAT?!" again.
Sirius's overall entrance is solid. In the first 3 books, Jo did a great job building up each one's subject (Philosopher's Stone, Chamber of Secrets, Sirius), so each of their entrances to the story is satisfyingly dramatic.
And finally, Harry's cementing of the connection between himself and his father. Every young boy wants to connect to his father. This connection that transcends life and death is locked together when Dumbledore tells Harry, "Prongs rode again last night." And when he tells him, "Your father is alive in you, Harry." And when he tells him, "In a way, you did see your father last night, Harry. You found him inside yourself.