Jackson Wyoming offers a lot of fly fishing opportunities in a relatively small area (by western standards). If camping is in the cards Gros Ventre Campground on the Gros Ventre river could serve as a base of operations, and you avoid the added fees and crowds of camping in the national park. But with the added fees of the national park you can add to the fishing opportunities so you may want to camp in Grand Teton. There are some campgrounds with absolutely breathtaking views. Not that Gros Ventre is lacking in that department either. If you go in August (late August is a really good time as flows are down and kids are going back to school and temps are downright nice) you can take advantage of a couple of areas that are only open in the fall to fish. Flat Creek on the National Elk Refuge offers some really good sight casting to very wary cutthroat, and Blacktail ponds offers the same in one of the most scenic areas I've ever fished. You could stop at the world famous Jack Dennis High Country Outfitters in downtown Jackson. Jackson is pretty touristy, but its a neat place to get a hat, and the guys at the shop are very knowledgeable. Remember sometimes in fly shops and bait shops you need to listen to what isn't said as much as what is said. The whole area is very researchable on the web and there are so many places to fish there, I think a week wouldn't cover half of them. Lots of good water with good public access. Guides are also available, and floats on the famous Snake River a possibility too. Its one of my favorite places to visit.
If you go the Duboise route be sure to check out Double Cabin Campground. It is an hour or so drive on gravel road and switchbacks from Duboise. There are 2 mountain creeks there and they are full of cutthroat, and brook trout, and a few rainbow. They have been having a heck of a time with the Pine bark beetle there, so the trees aren't so pretty right now, but the mountains are beautiful. The wading is pretty easy, and the fish are pretty eager. They aren't huge, but the last time I was there I was pleasantly surprised with the size of some of my catches, so there are some nice ones in there. Classic Mountain dry fly fishing at its best!