Fly-fishing is not difficult.  Unfortunately, this myth seems to be the main reason why people are reluctant to give it a try.  With the appropriate equipment and proper instructions, you should be able to cast the line reasonably well within a few hours.  Another misconception is that fly-fishing is just for trout.  This is simply not true.  You can fly-fish for bluegill, bass, pike, tarpon, stripers, or anything else that has fins.
Now that we cleared that up, let's jump right in and get our feet wet.  
When spin-fishing, you cast a lure attached to a very thin line with a spinning rod.  
The lure has weight and this loads the rod to propel it towards your target.  
The fishing line is just along for the ride.  

When fly-fishing, you cast a flyline attached to a leader and fly with a flyrod.  
The fly is almost weightless.  The leader it is attached to, which is usually
around 9 feet long,  is very similar to standard spinning line.  This is attached
to a flyline, which is usually about 90 feet long.  The flyline is made of a flexible
plastic and is much larger in diameter than spinning line and much heavier.  
This attaches to the flyrod, which is usually between 7 and 10 feet.  
When fly-casting, the flyline provides the weight to load the rod and propel
itself towards the target, with the leader and fly just along for the ride.  
It is very important to understand that you are casting the line, not the fly.  
The line and the rod have to be matched to each other in order to work properly.  

In spinfishing there is a large tolerance between what works and what doesn't.  
You could put 10 pound test on an ultra light and 6 pound test on a saltwater rod and they would both work.  Flyrods and flylines have to be matched carefully.
The whole purpose of all of this, besides the grace and beauty of it, is to cast almost weightless flies and present them in the most delicate manner.  It would
be impossible to cast most flies with any other kind of gear, and to match the
delicate presentations you can achieve with a flyrod would be just as impossible.  
This is why most people think of trout when they hear the word flyfishing.  

The flyrod is the tool of choice for most trout fisherman, especially in streams.  
Trout in streams need to be fooled with realistic imitations of their
usual diet.  And most of their diet consists of small stream insects.  So now
you might be wondering why you would want to use a flyrod on something
like a bass that likes a big meal?  The flyrod can still be more productive in
certain situations because of the delicate presentation, but most people do it
for the challenge and the joy they get from using such a marvelous tool.

Equipment

It is very important that you choose equipment that is well balanced and suited to the particular fishing situation.  While a particular flyrod may be used in different fishing situations, it's range of applications is far more narrow than that of a spinning rod.  You can take a 6 1/2 foot medium action spinning rod spooled with 8 lb. test and catch just about anything.  You could throw a small trout spinner or a large bass plug.  It may not be perfect, but it will work just fine.  With fly-fishing equipment, you have to choose the gear with a certain type of fishing in mind.  Again, there is some freedom in the equipment's range of applications, but it is not as flexible.  Well-balanced does not refer to the center of gravity, but the ability for all of the components to compliment each other and work as one.  There are six main components that compose a fly-fishing outfit:  fly, flyline, leader,
rod, backing, and reel.

Flies

Now it's time to choose what sizes and types of flies you will be using.  
There are two main categories of flies: flies that rest on the water (surface),
and flies that go below the surface (sub-surface).  The most common types of above surface flies are dry flies and poppers.  Dry flies are imitations of the
adult stage of small insects, and usually suspend themselves in the surface film
with the use of a feather collar or hackle.  These flies are practically weightless,
and are not too wind resistant.  Poppers can imitate anything from a frog to an injured baitfish, and are usually made of balsa, cork, foam, or spun deer hair.   These flies have much more weight and wind resistance to them, therefore they require much stronger tackle to cast them efficiently.  There are three main types
of sub-surface flies: nymphs, wet flies, and streamers.  Nymphs and wet flies
are very similar, they both represent insects in their aquatic life stage.  This stage comes before the adult stage (dry fly).  While nymphs and wet flies may
imitate slightly different things, the main difference is wet flies have wings and nymphs do not.  These flies weigh a little more than a dry fly, and weight is
often added to them in order for them to achieve the proper depth.  
This additional weight makes them a little harder to cast, but  the good news is
that there is almost no wind resistance.  The final group is the streamer.  
A streamer is usually tied to imitate a baitfish.  They are tied on longer hooks
and have long sloping wings to form the body of the fish.  They are usually
a little heavier than the nymphs, and the wind resistance can vary
depending on the particular fly.

Flyline

Flylines are classified by weight, taper, and density (if they float or sink).  Flylines are categorized by weight into a number system, which runs from number 1, which is the lightest, to number 15, which is the heaviest.  The lighter lines are more delicate in their presentation and they cast small flies well.  The heavier lines are less delicate in presentation, but provide the power to cast large, wind-resistant, and heavy flies.  Flylines in the 4 to 10 range are the most common.  Most trout fishing situations call for a line between 4 and 6.  For bass, line weights between 7 and 9 should be ideal.  Panfish rods fall between the trout and bass rod.  For saltwater anglers, you will probably want to be in the 8 to 10 range.   This chart may clarify things a little bit for you, but please keep in mind this chart is just a generalization.  It is to be used only as a rough guide.

Flyrods

        Now for the Rod.  Flyrods come in all lengths, weights, and materials.  Older rods were made of bamboo, and these are expensive collector's items today.  Some rods are made of fiberglass, and these are usually of poor quality.  They may say graphite on them, but the percentage used is minimal.  You can spot one of these rods right away.  
If you look at the diameter of the blank right above the grip, it is very large compared to the same weight rod in a more expensive true graphite model.  The most common material is graphite and this is the only one you should concern yourself with for now.  
It is much lighter than fiberglass, and also casts much better.  More or less, you get what you pay for when it comes to flyrods.  If you see a flyrod for thirty dollars, and it says graphite, you can bet it's really fiberglass.   True graphite rods start around fifty dollars and this should be your minimum.  The weight of a flyrod is the manufacturers suggestion as to which weight line it will cast the best.  Therefore, a 5 weight rod should use a 5 weight line.  Rods usually have a marking on the blank, just above the grip.  
It will tell you the length, weight, and sometimes the physical weight of the rod, the number of pieces it comes in, and the material it is made out of.  It should look something like this:  8'6" 5, or 865.  Both of these designate an eight and a half foot rod which should cast a five weight line.  It may also look like this G906, which is a nine foot rod for six weight line, and the G stands for graphite.  There are many different actions or bending properties for flyrods, but you don't have to worry yourself with that for now.  The other consideration is length.  A long rod generates more line speed, and it's length helps to keep your line from hitting the water or ground on your backcast.  A shorter rod is better suited to tight fishing conditions.  Say you are fishing a narrow stream lined with bushes and trees.  The shorter rod will be much easier to handle, and in a situation like that you will not have to make too many long casts anyway.

Reels

There are only a few different types of reels.   The most common being the single action, which is pictured here on the left.  With this type of reel, the handle is attached directly to the spool.  There are no gears to change the ratio.  These reels usually have a spring and pawl, or a disc drag.  The spring and pawl is a simple and inexpensive drag, and is satisfactory for most smaller species such as trout and panfish.  The disc drag is smoother and more precise.  This is the preferred drag system for bass and saltwater anglers, where you must control a very powerful fish.  Many reels have what is called an exposed rim.  This is a very important feature to have.  It allows the angler to apply drag directly to the spinning spool with the palm of their hand.  You should insist on this feature when purchasing a reel.  There are also reels that have gears to multiply your input, and there are reels that incorporate an anti-reverse handle.  These are nice things to have in certain situations, but they are specialized in their application range.  The last type of reel is an automatic reel.  This reel has a large spring inside to allow you to retrieve all of your line with just a push of a button.  These reels are not very common.  They are heavy and do not store enough line.  Your first reel should be a single action.  When shopping for a reel, you will want to see what the capacity for the particular reel is.  For example, reel X might hold a weight forward 5 weight line and 120 yards of backing, or a weight forward 6 weight and 80 yards of backing.  Therefore,
you should choose a reel that holds the line and amount of backing you chose earlier.

Conclusion

        Remember that flyrods are specialized, and one outfit can not do it all.  Therefore, you should choose an outfit that lies on the middle ground for the species you plan to pursue.   You can purchase the specialized outfits later.  Saltwater rods can vary greatly, but it should be at least 9 feet and
at least and 8 weight for general applications.

  Don't be intimidated by all of these variables.  Your local fly shop will be glad to help you set up a well balanced outfit.  You should know what the basic variables are so you can convey what you want the outfit to do, and you can understand what the salesperson is telling you.  A decent outfit should cost you between
$100 and $150.  If you can afford to spend more than this, by all means do so.  Remember, you get what you pay for and a quality outfit is something
you will get years of use and enjoyment out of.