Unfortunately, late May is not a great time for stream fishing out here, at least not what most people have in mind when they are headed out here to try a little fly fishing. That's getting into run-off quickly, and we are looking for a pretty healthy run-off this year, given the snowpack. But run-off is highly variable over it's course, tied to fluctuations in weather. Fishing can be good but location and technique are key. Late May and early June is actually the best time to be fly fishing the warm flatwater around here. That's what I'm doing then, when I'm not guiding people who came out to fish for high country cutts on Memorial Day.
Red Owl's advice about not worrying about distance is sound. Concentrate on good form, deliver what distance you can in two or a max of three false casts. You can't catch anything when your fly is in the air, and when you are making 10 false casts for a 10 foot drift, your fly is spending most of its time in the air.
Also, distance isn't as important in a lot of situations. A lot of times, what you need to do is be stealthy and get close enough to control the drift and make a good presentation, not dragging your flies unnaturally all over the place.
Keeping your casting stroke short, straight back from the target, straight towards it, rod tip high in the air -- 10 to 2, as they say, though that should rotate a bit forward for the shorter casts you're making as a beginning fly caster. Apply power (and a short cast with a trout rod doesn't take much, as a decent fly rod is good at translating energy to the line, if you stop the rod and allow it to), so stop the rod, let the power transfer from rod to line and let it roll off the tip. If you are presenting the fly on that forward casting stroke, then follow the line to the water with the tip of the rod. If you are not, holding the line lightly in your line hand, you will feel it tug if the rod properly loads and you can let it have some line. Concentrate on control of the cast over shooting a lot of line out, to begin with. You'll eventually start to feel the line loading the rod, and you should start to get a feel for cushioning the stop going forward and back so you don't shock the line and can cast with some touch.
Red Owl is also correct that if your back casts suck, so will your forward cast, at least until you learn how to recover a bad cast and then how to use a weak backcast intentionally (backed up against trees, for instance).
As the water comes up with run-off, a lot of places see clarity go down (less so at higher elevations, though in late May there are not yet many good high elevation options), and you can fish some pretty big flies. Stonefly nymphs, Woolly Buggers, all sorts of streamers. I like to fish pretty aggressively in the big water, smacking big streamers into foam mats and hole and such and ripping them out of there.
Reel with the opposite hand you cast with. A right-handed caster reels and tends the line with the left hand. There are fishing situations where this becomes important, as when needing to get a fish on the reel quickly and then when handling big fish close in on the reel, and switching hands with the rod is not what you want to be doing right then.
