Harry Potter and the Goblet of Fire
The transitional book in the series. Rowling still feels indebted to the "boy wizard-detective solves mystery" structure of the earlier three books, but she's also clearly grown more interested in character development and the long-term narrative elements of the story world. This creates an interesting tension: between the desire, on the one hand, to write another self-contained book (like the first three) and the desire, on the other, to write a book that functions as an installment in an ongoing serial. The tension isn't fruitfully resolved: this book is the slowest to get rolling (it takes nearly 200 pages just to get to Hogwarts) and Rowling's heart doesn't seem to be entirely in the mystery: it's the one of the first four which has the least satisfying Big Reveal, which requires an entire chapter's worth of flavorless talk to fully clarify.
Harry Potter and the Order of the Phoenix
The first book in the series where Rowling assumes that the readers have read the previous books. Freed of the necessity to ponderously re-establish the backstory-- the flaw that weighed down Goblet of Fire --Rowling is freed up to hit the ground running: the turbulence begins to hit with the first chapter. As with the earlier books in the series, the book is centered around a mystery here, although unlike the earlier books, it doesn't truly belong the the genre of The Mystery as such--there is no real way to puzzle out the solution, for instance. But the series doesn't really need to rely on mystery structure any longer anyway: by this point the long-form plot has amassed enough potential energy that it can soar simply by exploiting the conflicts already set up in its first four installments. Which isn't to say that there aren't new ones as well, notably in the form of Dolores Umbridge, whose petty abuses of power, disdain for the autonomy of young people, and Kafkaesque punishment schemes make her all-too-familiar: possibly my favorite villain in the series. Recommended.
Harry Potter and the Half-Blood Prince
Feels a bit like a book-length positioning of pieces for the big finale of Book Seven. Not that there's anything wrong with that: Rowling, at this point, has developed a very rich world, populated by literally dozens of characters who we care about, each with their own interesting plot arc. (This may form the basis of an entire later post.) Watching this network click forward in the standard increment (one year) is fascinating unto itself; the Voldemort backstory that forms the real backbone of this book is an added bonus.