Keeperjb- welcome to the forum-- you'll also want to check out the fly tying forums here too for lot's of info in past threads, especially about gear to get started, questions on materials etc. A lot of it can seem overwhelming at first, but hang in there. As you start tying you can also feel free to post pics of your flies if you want some constructive feedback from other tyers.
We also have fly swaps that come up from time to time so feel free to jump in on them as time allows--- all skill levels are welcome, including beginners.
OK, now for some specific suggestions--
Great advice from Ard-- Flick's book is a classic (and inexpensive).
Fishing during the summer can be tough at times. But at least in most mountain streams, fish tend to be a bit less picky--- the water tends to be moving faster which doesn't allow much time for close inspection, and the mountain streams tend not to be as productive as tailwaters and spring fed creeks, so fish tend to be more eager to grab a potential meal.
There's a lot of great resources out there on the web, including you tube vids etc. Charlie Craven's website www.charliesflybox.com
will be an excellent resource for you. BTW, Charlie's book "Basic Fly Tying" is the best one out there for someone just starting up tying trout flies, but you'll definitely get a feel for the quality of instruction from accessing the (free) info on his site which include step by step instructions for many patterns. To access the info click on the fly box link on the left of the main page. It will bring you to a drop down menu with "1 shop pictures" showing--- but click on the drop down and it will take you to a list of patterns. You can click on each one for a materials list and a step by step tutorial with pics. You can find step by steps to tie all of these on Charlie's site (as well as other sites on the web).
For a good start some woolly buggers, a couple of nymphs, a wet fly and some dries to imitate a caddis, a mayfly, and a terrestrial would be a good start to have a well rounded box- but as Ard suggested, for a mountain stream it would be pretty tough to beat an Elk Hair Caddis and an Adams just because they're so much fun to fish and to see a trout come up to the surface and whack them. Although they might be a bit difficult to tackle if they're your first flies. (The foam beetle might be a bit easier dry fly to tie, productive during the summer and still a ball to fish)
Here are some suggestions for specific patterns, a step by step tutorial for each can be found under the same name on Charlie's site. Going from easiest to hardest to tie:
Black or Olive Woolly Bugger- a good starter fly, this is also a good fly to use for fishing pools, ponds and lakes for trout and bass. A size 8 would be a good all around choice to start
Pheasant Tail (Nymph) size 16
Gold Ribbed Hare's Ear (Nymph) size 14
You can add some 2.4mm or 3/32" gold beads (these sizes fit most 14-16 size hooks) to some Gold Ribbed Hare's Ears and Pheasant Tail Nymphs.
Soft Hackle (a wingless wet fly). The body can be floss, dubbing or a "rope" of twisted peacock herl, with one or two turns of partridge or hen feather for a collar. Standard patterns include Partridge and Orange (or green or yellow) using silk or rayon floss or even tying thread, Partridge and Peacock, Partridge and Pheasant Tail, Hare's Ear and Partridge etc. All/any of these are a good searching flies and imitate various emerging, egg laying and drowned mayflies and caddis. Size 12 or 14
Foam Beetle- easy to tie, a good summer time pattern, doesn't use expensive dry fly hackle. You can get 2 mm foam at craft shops much less expensive than fly shops. Cut it into strips with a straight edge ruler and razor blade. You can use a sprig of white or yellow synthetic yarn in place of the yellow foam in the tutorial (easier) to help follow it on the water. Size 14
Elk Hair Caddis- a good high floating fly, and a must have in your box for mountain streams, size 14 and 16
Parachute Adams size 14 or 16 In addition to being an excellent all around pattern this will teach you to tie parachute style flies--- good mayfly imitations for slow water. By varying size and color you can imitate many different mayfly hatches using this "style" of tying.
There are many other options as well, but these are all good basic patterns that are good fish catchers in their own right and are proven patterns that catch trout everywhere they're found. And they're also are good patterns to learn because they'll teach you a range of different fly tying techniques used on zillions of other patterns.
Hope this helps--- keep asking questions--- and hope to see you in the fly tying forum. Good luck!