Hi, and welcome to the forum.
All you need to play the game is the Revised Box Set. All the rest of the books expand on this set of rules.
The Revised Box Set comes with a huge bundle of counters, so you can play the game straight out of the box. And test the many fleets to get the feel of the way they play and an idea of what minatures you may want to buy, in addition to the ones that look really cool.
Sky Full of Stars, expands on the rules by adding, amongst other things, Space Stations, Scout ships, boarding actions and more detailed planetary assault rules. It also massively improves the fleet lists available and expands on the campaign rules. I would highly recomend this book if you like the game.
Armageddon, expands the game by adding the Armageddon level ships, splitting the Earth Alliance fleet into 3 distinct eras, revamping the Vorlons and Shadows, adding the Ancients and tweaking the fighter and Stealth rules. This book is not necessary unless any of the updates would be of interest or use to you.
Dilgar or Drakh fleet books, provides the full fleet list and necessary rules for using the Dilgar or Drakh as a race in the game.
Earth-Minbari war, provides a bunch of campaign scenario's and some special characters for the Earth Alliance and Minbari.