I live in Montana but pretty close to Wyoming. I would look at the Bighorn in Wyoming. I would look at both Teton National Park and Yellowstone. Numerous rivers to fish(Snake, Gallatin, Madison, Teton, Yellowstone, Gibbons, Firehole, Lamar, etc...)in both of those places. I would tell you to come a bit later than August. September and October are special times of the year up here for both wildlife and fishing. October is off the charts on the Madison in the Park. Fish are coming out of lake to spawn and you can catch incredible quantities of big fish.
Crowds are less in October and wildlife is better viewing. If you came in September the elk are in rut and that is really cool. We moved here from Texas 6 years ago and I still own a ranch there but you should give yourself as much time as you can on your first trip up here. It is a special place.
I would suggest flying into Jackson and spending a couple of days there and then driving through park and staying in West Yellowstone. Madison is going to be on the Montana side some and some in Wyoming. The other idea is to do Flaming Gorge and Green River then drive up. I would suggest doing a few days in multiple places the first time. It is a big area that you could spend a month in and barely scratch the surface.
Let me know if I can help and bring bear spray!