Oh dude, I could write the longest reply in bluelight history on this subject..
I've spent hundreds of hours fly fishing. From the Upper Peninsula in Michigan, to Saskatchewan, and all over Ohio and it's great Lake Erie run-offs, it's really an enjoyable hobby everyone can take part in.
Rods:
A quality graphite rod is always your top choice. Start off with a pole no longer than 9 feet 6 inches in length. You can work your way up to a bigger, longer pole as you become comfortable with the way it works. 9.5 feet gives you a good feel for using something longer than a traditional fishing rod.
I personally use a 12.5 foot rod now, it actually breaks down into 3 pieces so I can store it in my trunk.
Reel:
People will tell you that 'you get what you pay for' alot of times, but I don't think this is the case with fly reels. I've had just the same luck and quality with a 'beginner' cheapy reel than I do with my current one ($160) Make sure when you buy a reel you get one with a large spool capacity, and an open face reel.
Open Face vs. Closed
This gives the wet line a chance to dry, along with the inner workings of the reel in case it goes underwater. Plus, it lets you see how much line you have remaining - crucial when a big fish rips off 60 yds. of line in 3 seconds. You can find fly reels priced from $30 - $350+ so it all depends on what you want to spend.
Line:
For short to medium casting range situations, stick with a Double Taper line as opposed to Weight Forward. If you are consistently throwing longer casts, you can make them with fewer false casts with a Weight Forward line. For a beginner, unless you're going after Tarpin in the deep ocean, a DT line is ideal.
At the end of my "regular" tapered line I use a mono-filament leader of 6-8 feet. Fish tend to spook easier when they see a white of neon green fly line slap on the water surface. The clear mono leader gives you some room to present your fly free from distraction.
Bait:
I could never tie a fly correctly to save my life, maybe you can. People make that their hobby - and to those 7 people in Arkansas who find fly tying enjoyable I salute you! But for price and convenience factor, any rod and reel shop offers dozens and dozens of flies ready for purchase that will suit you perfectly. The times I did tie my own, the ones that I meant to float sunk, and the ones I wanted to sink floated.

Remember to match the hatch, don't go throwing bumble bee flies on a river when the mayflies aren't even hatched yet.
Gear:
Don't skimp on 2 things - Waders and polarized sunglasses. The waders are an obvious choice, here you DO pay for what you get. Try then on before you buy them, so you can feel how they fit and where the rubber seems are (sometimes they rub you raw if they're in the crotch area). Some people prefer the boot to be a part of the wader itself (I do) others don't. Remember to hang them upside down when you are done with them. If you fold them up and stow them away, the rubber cracks at the folds and you'll be stuck with wet feet the next time out.
Polarized glasses let you see the fish in the water, better. They really do cut down on the glare so you can focus your casting. I use a $15 pair I got at Walmart.
Casting:
There's no possible way I could describe 'how to' through this medium. You can watch it on TV and get an idea of what you're supposed to be doing and how you should look. It's important to remember to keep your rod at a nice even angle when casting. Try to pull the line back and let it go with your casting arm at the 2 o'clock position.
Watch out for obstructions behind you! I've snagged my line on a tree that's been 25 feet
behind me because I was making such a long cast.
Try
this site and click on
Fly Fishing Basics for a good run-down on everything. Good luck!