Here's more than you probably want to know, but it's specific to Washington mountain fishing. If you're looking for weekend, car-based fishing, you can ignore it. If you're looking to add to a BACKPACKING experience, I think it's pretty solid.
You probably don't want to go with fly fishing only. That might not be a popular statement and I'm sure some will disagree, but in Washington's mountains (where I do 99% of my backpacking) you will catch a lot of vegetation and not a lot of fish. Washington's mountain lakes are so lush and often in steep bowls and you have either cliffs or trees or high bushes on virtually all shoreline in the mountains. You will have a very frustrating time since it's nearly impossible for even modified backcasting. Pretty much every high lake fly fisherman I know of uses a raft or float of some type. That's a much larger commitment of gear and weight. Absolutely a blast if that's what you're into, but it sounds like you don't want to make the fishing the center of your trips, right?
That said, I do believe flies are a superior way to fish and nothing frustrates me more than catching a fish that has a bunch of neon, glitter-covered playdough in its belly.
There is a high lake hybrid setup that basically uses a fly rod, flies, but a spin reel and "bubble" that can be filled with water. This setup was dialed in by a bunch of Rocky mountain fisherman in the 60s that were frustrated because they could see fish rising and feeding but they were always out of range. The setup is built for incredibly long casts. I can assure you that if done right and once you get the hang of it, it's amazing how far you can cast a fly with this setup. I am by no means the world's greatest fisherman, but I can honestly say that I have NEVER been fishing with someone who can even come close to the distances I'm casting. Typically they have trouble getting even half of what I am, and that's with a regular spin setup. This is important because you might only have a few square yards of standing space to fish from and unless the water you want to reach is right in front of you (rare), you need distance to get to fishable water.
Check Amazon or eBay for a book called "Fish Don't Think". It's cheap and is probably the best book on this setup that exists. It's got a lot of great info for applicable fly fishing mountain lakes as well. The setup is light, too. I use a 2.9oz rod, a reel that's a bit under 6oz. Add some flies, a couple bubbles and as much extra stuff as you want and you have a setup that will rival most fly setups for weight. Grab a 3-4oz. fly reel and if you do happen to find magic water you can have the best of both worlds for a pretty small weight penalty.
I really don't think I have any special fishing powers, but I fish almost every trip I go on and in the past 3-4 years I have not been on a mountain lake where I have been out-fished by a friend or anyone else on the same lake. I don't want to sound competitive about it, because I actually like the super mellowness of fishing high lakes, but just as an observation, it's pretty obvious this setup is uniquely tailored to mountain lake fishing. It'll get the flies to the places the fish are – you just have to get the right fly and get them in!
Happy to answer any specific questions. Definitely try something out – it really adds to the backcountry experience!
-Curt