You should get a DT6F or a WF6F; beginners find it easier to learn to cast with a slightly heavier line, because it's easier to feel the line load the rod.
The leaders are tapered strands of monofilament. The taper serves as a continuation of the taper in the end of the fly line: it allows the energy stored in the fat "belly" of the fly line to disperse gradually as the line and leader roll out over the water; otherwise, the fly and leader would dump in a heap. A nine foot, 5X leader will work well for most dry fly trout fishing. You should also buy a little spool of 5X material, so that you can replace the tippet (small end) of the leader as you use and change flies. Otherwise, you'd soon have a short, heavy leader that wouldn't fool trout, and might not even fit through a hook eye. The leader will be kinked from the package, and probably every time you pull it off the reel. Stretch the leader (and any kinks in the fly line) between your hands for about five seconds.
Before your trip, practice casting with a little tuft of bright yarn or a feather in place of a fly. When you can drop your yarn within a foot or two of a target from 20 to 30 feet most of the time, you're ready to fish.
A good instructor can teach you to cast competently in less than 30 minutes. If you teach yourself, there'll be more error than trial, and it may take years to unlearn bad habits. Although it's hard to learn it all from secondary sources, they help: illustrated books and articles, videos. Fly casting is much more about timing than power.
There are thousands of fly patterns, and more being invented all the time. No, you don't need most of them, but you do need some variety. The reason is this: trout usually want what they want when they want it. If at 11 a.m. on the part of the stream where you're fishing, a hatch of 1/4" long rusty brown mayflies are emerging through the stream's surface, you just about have to have a rusty brown-body dry fly of that size, or you're going to get nothing but exercise and frustration. In some ways, it's easier to catch trout between hatches, by using an all-around searching fly like an Adams or Elkhair Caddis; that is, if the trout are a little hungry and are paying attention to the surface. Sometimes they just sulk for hours, and nobody can catch them. Trout, like most fish, eat a lot more near the bottom than off the surface; we just don't see it. That is why fishing a sunken nymph (pre-adult insect) under a bright little plastic or yarn indicator (really, a mini-bobber) at the upper end or middle of your leader is the default method for trout fishermen everywhere. It doesn't always work, but it probably works more consistantly than anything else.
The Gold Ribbed Hare's Ear and the Pheasant Tail, same sizes as adult insects, can work anytime.
If you've learned anything useful from the few paragraphs above, think what reading one good book about fly/trout fishing will do for you!