Like most of the others have said, avoid the cheap all-in-one fly tying kits. They will do one of two things; either the hooks will slip in the vise and you'll end up becoming frustrated and giving it up all together, or else you will become frustrated and end up buying higher quality stuff, thus wasting the money you spent on the kit.
In my opinion, a good vise is a must. That doesn't mean you need a $300 vise. There are plenty of very well made vises available for under $200 - Renzetti, Dan-Vise, Dyna-King and HMH off the top of my head (my personal favorite is the HMH Standard but lots of people like true-rotary vises). The vise is probably the most important fly tying tool you will purchase and the only way to get something that works for you is to tie on them. Either find a friend that has a model you're interested in and tie a few flies on it or else go to a shop. Most shops will be happy to let you tie a few flys on different vise models.
The second most important thing is scissors. Don't skimp on the scissors either. Dr. Slick make some really great scissors for $20 or less. Personally I favor the Wiess Quick-Snips which are also available for under $20, but have a look at scissors from different manufacturers and see what you prefer. Another piece of advice, if you can afford it buy two pairs of scissors - one pair for cutting delicate stuff (thread, etc.) and one pair for cutting tougher stuff that will quickly dull your "good" scissors (tinsel, wire, hair).
In my opinion, those are the two most important things - vise and scissors. What tools you'll need other than that will depend mostly on what type of flies you're interested in tying.
As far as materials, I have to echo what has already been said here. Buy the materials that you'll need to tye one or two of the patterns you'll be fishing most. Perhaps an Adams and a Hare's Ear Nymph for trout. After that, simply continue to purchase what you need to tye the patterns you want to tye and you'll quickly amass a great variety of materials.
Good luck and I hope you enjoy it. I love tying every bit as much as I love actually fishing. If done properly, they both give vast amounts of enjoyment for a lifetime. If you're anything like the rest of us, you'll begin haunting the craft stores and staring yearningly at roadkill you pass on the highway! Enjoy!
-Darryl