Fly-fishing is a very traditional, relaxing sport which consists of getting a fish to eat a hand-tied lure, or fly. It can be extremely rewarding and interesting activity that I hope you will try.

Step 1: What Fly-fishing Is

Fly-fishing(sometimes called the quiet sport) is the tricking of a fish to attempt to eat an artificial(no live bait...ever) fly that you put out to for it.
The there is a big, obvious difference between fly-fishing and what is known as in-line fishing. In in-line casting the lure pulls the line to the fish, while in fly-casting the line pulls the lure.

Step 2: Basic Equipment

There are some things that are always necessary(sorry bad camera)

1.fly rod
2.fly reel
3.line(thick colored stuff)
4.leader(tapered nylon)
5.tippet(un-tapered mono)
6.fly-box
7.flies(i tie my own)
8.dry-fly floatant
9.at least one knife
10.fish stringer(if keeping fish)
11.liscense(you better not poach)

Step 3: Rod and Reel

The most important thing here is balance, which is incredibly easy with the right information; a / means it can be either of the two weights.
The weight for a rod is dependent upon where you are fishing and what you are fishing for:
3/4 or lower = small pond, bluegill
5/6 = most everything and everywhere not in the extremes of small and large, begginers rod
7/8 = large lakes or rivers, larger bass and trout
9/10 or bigger = don't go here without experience these will go from rods for 10-pound fish to 2-handed (spey) rods.
Choose a reel with the same weight as the rod, this is balancing. There are three main type of reels:
single turn = one turn of handle is one rotation of the spool,basic
reels with gears = one handle turn is about 2.5 turns of the spool, a little more complicated and expensive
auto-reel = wind then push release button, should definitely not start off with this type of reel(slight exageration)

Step 4: Line, Leader, Tippet, and Backing

Sorry for the fuzzy picture of a new bass leader, bad camera.
Backing gives you extra room to play with the fish. You can add backing after the line to find the right amount then re-assemble the setup, but 100 ft. is usually enough.
Lines are the thick, obvious, cords fly-fishers have flying over their heads this is the muscle that clealy transfers energy from your rod along to your fly; color does not really matter; a taper at the end helps with casting; I recomend double tapered line.
Leaders are a tapered length of monofilament or braided nylon line that fish can't see and make the fly hit the water first. Choosing these are easy, look at the printed information, and the pictures can be a decent guide to choosing one too.
Tippet is mostly for maintaining tippet length. This must be as thin as the smallest point of the leader. Just tie a foot or two onto a short leader.

Step 5: Types of Flies

The universal concept of match the hatch applies; choose a fly that looks like it could live in the water you are fishing. Though you sometimes want something gaudy to and or intrigue the fish. You can always play "Musical Flies",and pick one that looks cool or you think could live in the body of water you are fishing in. Bigger is not always better.
There are 5 main types of Flies:

Dry flies - float on top of water, you must apply floatant on all but foam

Nymphs - look like the aquatic larval stage of bugs

Wet flies - look like drowned flying bugs, very traditional

Streamers - have a long feather for a wing or tail

Hairwings - like streamers, with hair replacing the feather

Picture is of a typical bottle of floatant.

Step 6: Assemble Gear

To put lines and reel together do this; knot are common and easy to look up.

Backing to reel: arbor knot or whatever will hold

Line to backing: albright knot or loop-to-loop connection

Leader to line: nail knot (important)

Tippet to leader: blood or surgeon's knot

Fly to tippet: Trilene, orvis, or whatever knot you use to tie onto swivels when you have gone in-line fishing

Step 7: Choose at Least One Good Knife to Bring

Pretty basic here, choose a knife that you can gut and cut lines with. I usually take at least three.

Step 8: Casting

Basic hand duties:

good hand = control rod, fingers put tension of line as it passes though them

bad hand = control rod, strips and pulls line

There are differing opinions here; it is a matter of preference. First you usually swing the rod back and forth letting line out with you bad hand, this is called the false casts. When you have enough line above your head you bring your hands together and throw your rod to 9:00. That should place you line and fly strait in front of you. These are very basic concepts. I am not there to show you so figure it out.

Most common method of basic cast: 10:00 to 2:00, more arm movement than 2nd method

Faster, brisker method: 11:00 to 1:00, bit more wrist

Experiment or ask for the help of someone who knows more.
This can be tricky to figure out. If the line lays out strait and gently you are doing something right

No good casting picture, bear with me

Step 9: Presenting and Retrieving Your Flies

After you pick a fly and start the casting process, you need to do two things to trick the fish: get in a good postion and make the fly move realisticly.
Cast the fly where it will go over/by the fish when either you or the current pulls it. What I mean by pulled towards you is that you pull the line that should be out on the water towards you; use your left hand to pull the line, with it sliding through the fingers on your right hand. Pull, or retrieve, in long steady strips; short quick ones; or a combination of both. The point here is to make the fly look real and delicious. Get familiar with the creatures you are mimicking if you are confused.

Step 10: Hooking and Fighting

If you see or feel a strike on your fly, tightly hold your line in your non-rod hand and quickly bring up the rods tip. If the fish keeps pulling after this, the hook is now stuck in the fish's lip. This is tricky.
To fight, or bring in, the fish just keep your rod held up and firmly use your bad hand to pull in the line and fish.
If you succeed you have successfully fly-fished. Remember, always try to experiment and ask for help if possible. I hope you try this, good luck.