There is an AFF Quickstart available in pdf format that provides an introductory adventure, basic rules and pregenerated characters. This is a fast and easy way to get into AFF even if you are totally new to roleplaying games.
Alternatively, you can purchase the Advanced Fighting Fantasy 2nd Edition rulebook. This also contains the Quickstart adventure but also has the complete rules that will allow you to play for many years. It contains a basic guide to the world of Titan and a handful of sample monsters to challenge the Heroes with.
In addition to this book, you might also want to consider the Titan and Out of the Pit books to provide an in depth description of the game world and 250 classic monsters respectively. As an alternative, the AFF Deluxe hardback book contains both thr rule book, Titan and Out of the Pit in one cover.
Once you have learned the rules of AFF (or maybe you already know them and want to get going with a more comprehensive set of books) you can then consider the wide range of additional books available. Note that none of these are "essential" to play the game, although the do all add something to the experience.
The Combat Companion and Heroes Companion are not large books, but they both contain a lot of extra optional rules. Adding these books into your campaign will offer new options to magic using Heroes and warriors of all types as well as make life easier for the Director.
Beyond the Pit and Out of the Pit offer 250 more monsters in each volume, and when added to Out of the Pit offer a huge choice of enemies (and indeed friends) for the Heroes to encounter.
Blacksand, Travels in Arion and Creatures of Mishna are books that provide a lot more information about a specific part of Titan. They make it much easier for a Dirctor to set (or even just start) adventures in that particular place, but also help give a feel for what Titan is like.
Adventures such as The Crown of Kings or The Citadel of Chaos are a very easy way to get going with a longer campaign. They offer an adventure structure that can be used as-is, or can be expanded and built upon to provide a much longer set of adventures.
Lastly, it is also worth mentioniing The Warlock Returns, the AFF fanzine. Packed with all sorts of information, rules, settings and other titbits, it is a way to make AFF accessible to Directors and Players of all experiences.