Lost Creek is nice- it's about an hour from my house- but it isn't classically "Rockies" if you know what I mean. Yes, it'll have little snow, but you won't see alpine-looking mountains. You certainly can fish in the various little streams though- I recall seeing some huge cuthroats (or cutbows?) in the Lost Creek itself.
If you want to climb mountains then, as others have said, you really need to go a little later in the summer. I hiked over Music Pass in the Sangres one June and the postholing was just barely bearable. Still, my pants were soaked to the knees.
Or, maybe bring snowshoes?