In order
1) Corebook- For any character.
2) The book for your character type:
Augmentation for Street Sams, Street Magic for magical characters, Unwired for hackers/technomancers.
3) Arsenal- For when you are comfortable enough that you feel like tweaking your guns, vehicles, and armours in exotic ways,
and having more exotic weapons.
4) Runner's Companion- For the extra qualities. Seriously, some of them were things that my group had been trying to
figure out before RC came out(Amusingly: I house ruled a Bi-lingual quality...and it matched EXACTLY what RC had)
For the Advanced Lifestyle Rules.
5) Attitude: Just the first two chapters ALONE make it worth it for a new player. They basicly talk all about runner teams and running, and give you a good feel for the world. I am putting it down here only because it is not a core book.
6) 6th World Almanac. Learning the history of the setting in far greater detail then the chapters in the corebook.
7) Special Interest Books
War! if you want the fun mil-spec gear, Spy Games if you want the espionage stuff(which, BTW, is fun as heck on a hacker),
Vice, for if you want to know about organized crime.
Safehouses e-book(to make CanRay happy....)
9) Everything else.