With this somewhat longer novel, Rowling appears to begin her Big Bad arc in earnest, invoking the history of the Potter parents' generation, and introducing more of what (I presume) will be major participants in the obviously forthcoming battle between good and evil. That being the case, the beginning of the novel is a curiously uneasy mixture of real anxiety (we discover, or think we do, that Harry is in genuine danger from an escapee from the wizard's prison, Azkaban) and campy humour (when he runs away from home, fearing punishment because he inadvertently inflated an infuriating Muggle relative, he ends up on "the Knight Bus" - a comical, badly driven, public transit for stranded wizards). The aura of dread increases with Harry's subsequent introduction to the Azkaban guards, faceless, soul-sucking, depression-inducing creatures called Dementors. In between those two things, Harry gets his first real experience of ordinary, loving family life in the home of the Weasleys.
I won't get into the particulars of the plot, but it is very much concerned with father figures (Sirius Black, the godfather, and Harry's relationship with his real, murdered father). Other memorable elements of the fantasy machinery are transformation into animals (Lupin, the new Professor of the Dark Arts, is a werewolf, and in his schooldays he had a small group of friends who also developed the ability to transform), and time travel (it's not just useful for attending multiple classes at the same time, as Hermione discovers).
Hagrid the gamekeeper again plays a major role, and we cannot help but sympathize with his great grief over the unjust execution proposed - and, we believe, carried out - on a rather splendid hippogriph (bird-horse hybrid) in his care, at the instigation of the ever-contemptible Malfoys, father and son. This narrative of false imprisonment and unjust punishment is, of course, a mirror of the main Sirius Black plot. Both of them escape their dire ends, but must nonetheless still flee the unjust. Another animal character, Hermione's cat Crookshanks, also gets a vindication of sorts from a false charge of "murder" (of Ron's rat pet Scabbers, who turns out not to be neither a murder victim nor a rat).
Once again, Dumbledore enables the juvenile heroes to carry out their virtuous quests, rather than exerting his own powers.
So far in the series, I get the sense that Rowling is very much in charge of plot strands, themes and characterizations, while still maintaining her sense of fun. Looking at the increasing length of the books in the rest of the series, I hope that the increasing complexity that length portends does not mean that any of those elements suffers. Can't wait to find out!