Last year most stuff in the mountains was still melting out in early June. It melts out fast, but if you to backpack, see the good stuff, have good fishing (ie not murky), and avoid postholing and so forth, July or August (or September) are better choices.
As for flying in somewhere and getting out into the woods and back without renting a car, I'm not sure how you'd do that. Public transit and shuttles really don't exist out here.
As for trips:
There have been several threads on Glacier recently. Tons of good options. Not the most fishing-centric local.
The Bob is amazing. After Le Parcour I want to get back into the central east side and do some fishing, and see the China Wall area without floundering on snowshoes.
A great scenic and fishing trip in Yellowstone would be to hit up Shoshone Lake, the upper reaches of the Bechler River, and the Upper Firehole. Thermal features, lakes, meadows, streams, woods, and wonderful, engaging small stream fishing for some very smart little trout.
Oh, and keep us posted on your intentions and plans. I (we) might be able to help out. I'm not sure what my plans will be after graduation in May.