If you tie, I would get the Flytyer's Benchside Reference. It's a huge book illustrating all the tying techniques known to man.
A DVD you should own and watch every spring is Joan Wulff's Dynamics of Flycasting. The book is good too, but I think a DVD is really the right media for flycasting demos.
Among the books, I'll echo some of rrt's picks
A River Runs Through It (the book is so much better than the movie, although I like the movie awfully well too)
The River Why
Anything by Traver
Anything by Gierach (but not too much at anyone time, he starts to sound like he's repeating himself after a while)
Anything by Rosenbauer, but especially Reading Trout Streams and Prospecting for Trout
An Angler's Guide to Aquatic Insects and Their Imitations
Landis's Guide to PA Trout Streams (I don't care for Meck or Wolf's) plus the Delorme. However, the FBC website has all that information for free too. You just can't dogear pages.
Harry Murray's Flyfishing for Smallmouth Bass
While we are talking about smallmouth bass, Clouser's Flies is excellent. And having his pattern for drakes is worth the cost of the whole book. Only pattern I've actually done well with on the Penn's Creek Green Drake hatch.
Bluegill Flyfishing and Flies by Roxanne and Terry Wilson. It'll get you fired up to find a pond with gills the size of dinner plates.
Either of Marinaro's books could be considered must haves, but I like In the Ring of the Rise more than A Modern Dry Fly Code.
TroubleShooting the Cast by the late great Jaworski. Don't pick up a long rod without it!
Either Joe Humphrey's Trout Tactics or On the Stream with Joe Humphrey's but not both. And don't take either too seriously or you'll think flyfishing is too hard for mere mortals and give it up entirely.
Production Fly Tying by A.K. Best. I didn't become a faster tyer after reading it, but my flies got a lot better