I'd like to call myself super-fan, but I'm not one to go around self-proclaiming, so yes, I'm a fan :3
You are correct. The original Avatar series, Avatar: The Last Airbender, was broken into three "books," or seasons, based on the element Aang was currently learning.
Book One is water, and the season finale ends in the... 4 (I think) part season finale, Siege of the Northern Water Tribe.
Book Two is Earth, and I forget the name of its finale, but I remember the episodes well and believe it was also four-part.
Book Three is Fire, and its season finale is Sozin's Comet, parts 1-4.
As Aang, the Avatar, already knew Airbending, there is no air book, however, it is important to point out that the first Book/season of The Legend of Korra is called Air.