Basic Definitions
Dry fly, these float on the surface, imitating insects such as flies, beetles, and moths.

Wet Fly, these are used below the surface and imitate things such as small fish, mudeyes(dragon fly larvae), leaches, frogs, tadpoles, caddis.

A dry fly requires a floating line, while a wet fly can be fished on both floating and sinking line, although a sinking line is more common. Anyone starting out should begin with a floating line.

The leader is a tapering piece of norman monofilament line, 5 to 10 feet long. It connects to the fly line at one end, and the fly at the other. These can be made cheaply at home by tying together between three to eight lengths of line, of decreasing diameter.

Backing can be placed onto the real before the fly line, as most fly lines are only about 20-30 meters long, so if you hook a big fish the backing can be played out like normal fishing line. This can be purpose bought, or simply use some old, strong monofilament. It is not essential when targeting small to medium fish.

What's that number 7 doing on my rod?
Fly rods come in different 'weights', this does not refer to the actual weight of the rod, rather the weight of the line it can cast. A 4 weight rod would be used on delicate streams, while an 11 weight rod would be used for long casts on lakes, or salt water fly fishing. Beginners should start with a 6 or 7 weight system. A 7 weight rod would require 7 weight line, and so on. If you already have a rod, get its weight rating checked before you buy line for it.

Fly reels vary a lot in price, the one I use is a plastic twenty dollar model I bought years ago, if you feel like it you can pay hundreds of dollars for a reel. Probably the simplest way to get your gear is to buy a 'combo' from K mart. These include rod, reel, line, and even a few flies. From memory they cost about ( on special) which is cheaper than if you bought it all separately.

The first Flies you should by would be a few red tags(dry fly) and for wet flies try a green matuka, wooly bugger, and mrs simpson.

You cannot learn to fly cast from reading a book, ask someone to show you the technique and closely watch other fly casters next time you go fishing.