The long and short of it is that your "must have" books vary based on the gaming group and what aspects of Shadowrun they tend to enjoy.
SR4A (20th Anniversary Edition), or at least the older core SR4 book, are cold, hard, must haves. It's hard to go wrong with Runner Companion, too, but you can get by with just the basic book if money's tight. At any rate, you -- obviously -- need the basic rulebook to jump in. Everyone does, no matter what "flavor" of game they're into.
Setting wise? If you're planning on the traditional Seattle sprawl, too, Seattle 2072 is a great book. Depending on the flavor of the game, Vice is good stuff, too. There's also Corporate Enclaves, Running Wild, and Feral Cities, that could all be very very helpful, or could be dead weight, just depending on what you, the GM, have in mind.
A more combat oriented game, with a group that's big on street sammies, mercs, and run-and-gun? Fans of The Transporter or Heat or Collateral? Augmentation and Arsenal will be important, along with Street Magic if anyone wants a cool Adept. Arsenal's probably the most "must have" of these, because it's also the closest thing to a Rigger Black Book so far for SR4, with lots of vehicles and mods -- so someone can have fun with a Wheelman type -- and you might also want Vice depending on what sort of jobs and contacts you've got in mind.
A group with more of a fantasy background, that's likely to be full of guys who want to play folks slinging mojo and wading in with esoteric martial arts and weapon foci? Street Magic, Street Magic, Street Magic. Lots of stuff in this one for the feel and flavor of magic, not just crunch (though the crunch is quite nice).
You and your buddies techheads? Eager to dive into the Matrix? Unwired and Emergence will be great, and probably some Augmentation, too (for high tech goodies).
Etc, etc. The "must have" books aren't "must haves" for everyone but us purists and collector's, really. A lot of what you'll "need" to run a game will vary based on the game you want to run, and your players want to play in. The basic truth is you can get probably by with just the basic rules if you want to, because it's a strong enough book to shoulder a campaign if you let it -- or, at least it can do so until you and your buddies talk after a game and someone says "Man, I wish I had more magic tricks" or "Oooh, new chrome would be awesome!"