I gave a full list because he indicated he'd previously been into SR. I left out critters because I'm not a huge fan of things that come with GM screens. I've bought several and never been impressed by any of them (Critters included). It
is good to have if you're wanting to use non-human threats, but, frankly, you could run a whole campaign and never use it. It's also GM specific so it gives less bang-for-the-book than something that provides GM and player material alike. If the group is going to spread out book buying amongst the players as well, I'd probably make it a GM priority. On a general scale, there's other stuff which gets used a lot more in most games.
Additional notes:
- SR3 Companion: This book is strictly optional. It provides additional ways to create characters and things like edges and flaws. It provides more example contacts and things like contact upkeep and game balance stuff. If you want to keep things simple, you might not want to allow PCs to create characters using it. It has a lot of advice and stuff for GMs which you might find useful, though.
- Man and Machine: Because of the rules for chemicals and drugs, this book is handy for the GM as well as players. There are a lot of cool toys for your street sammies and even mages and adepts who don't mind loosing a bit of magic. And anyone who likes not to be seen will drool over a ruethinium cloak. Additional grades of cyberware and bioware provide the street sammie's advancement path.
- Cannon Companion: This book has a wide variety of weaponry inside as well as things like martial arts. It's pretty useful for every character type since it allows you to tweak your arsenal with things like custom grips and other weapon modifications. There are also handy armors and armor modifications (including high-fashion armors).
- Magic in the Shadows: This book provides magical advancemnt paths (initiation) for your adepts and shaman/mage characters as well as many magical threats (bug spirits, blood spirits, toxic spirits). There are also additional spells and adept powers which allow for more character variety.
- Rigger 3: Serious vehicle characters pretty much require this book. You've got just about every vehicle you're ever going to need, rules on how to do various things with electronic warfare and rigging, and ways to tweak your ride. Even non-rigger characters do well to have access to the expanded vehicle list so not everyone is tooling around in the same type of vehicle.
- Matrix: I've heard it said that you can't really run deckers without this book. I'm not too sure that's true, but it does provide the decker advancement path (programming your own stuff, since buying it is insanely expensive at higher ratings) along with custom decks and the like.
Whew. Okay, you're trying to get your group into SR. I'd go get the main book (aka, the Big Black Book -- BBB or SR3) and toss everyone an archetype (the gear's in the back of the book, by the way, the characters are on the big color sheets). Play a quick game or module (First Run works well) and let everyone make their own characters after that if they want. I know my first SR game was with the troll combat mage out of the BBB and I rather liked him

For buying the books, priorities should depend on characters and budget. If no one is playing a magic type, you can skip MitS. If no one is playing a decker, skip Matrix, etc. Some of them are useful no matter what characters are at the table and some are strictly optional (SR3 Companion fits this bill... PCs tend to use the first sections which apply to character generation and GMs tend to use the rest of the book, if at all.).
If I'm working on characters, I generally drag around 4 books with me. The Main book, Cannon Companion, and:
- Sammies: Man and Machine and Rigger 3 (usually)
- Deckers: Man and Machine and Matrix
- Adepts: Magic in the Shadows and Man and Machine (stealth cloak and drugs)
- Magic-types: Magic in the Shadows and... okay, just 3 for mages.
The Companion might come along if I need to work with edges and flaws a lot, but if you have one copy in the group, that's probably enough. I also frequently use the Sprawl Survival Guide's expanded lifestyle stuff. I'd advise getting it if you're not that fresh on the setting or you've already covered you basic rules needs. I don't know about it as a primary priority unless you're really in need of setting material. Setting books are also really useful, like New Seattle (which, incidently, I own two copies of... because I'm an idiot and lost (but later found) my original copy) or Shadows of North America.
Why, yes, I do like the sound of my own typing. Why do you ask?