I've been there once, and I fished the park some. It's mostly wild bows, by the way, but there are some brookies and browns as well. Most of the streams in the park are going to have fish.
Generally the fishing was similar to smallish freestoners here. Aggressive smallish fish and not much bug life. Attractor dries worked well. If you can't coax em to the surface that day, feel free to run a dropper. Water was crystal clear and the fish spooky, but not picky, based on my experiences.
Anyway, it wasn't a fishing trip, and my wife was heavily pregnant at the time. I got a number of short outings in over a week while she did some touristy things, but due to the pregnancy we were limited on how "adventurous" we could be. This was mid-July. List them here:
Cosby Creek, around the campground and above. One of my favorites. Lots of brookies and a few bows. Fished it a couple of times as I was camped there for half the week.
Roaring Fork, which parallels the Roaring Fork Motor Trail right out of Gatlinburg. I only fished it for like an hour as my wife was doing touristy stuff, but it fished well. I was in double digits in that hour. All bows.
Little River - above the Elkmont Campground. This was my longest outing. A few bows and one nice brown. Bigger water.
Little River - in the town of Townsend (outside of park). Caught a few bass and had a big stockie bow to my feet. It's big water down there.
Middle Prong Little River - Just below the Middle Prong Trailhead. Caught a number of bows here. Was just after a T-storm, so I may have lucked out and got them when they were active.
Little Pigeon River - I didn't do well here. But I was on the lower end, just inside the park and along the road, and it was mid afternoon.
As for fly shops, Little River Outfitters in Townsend TN was a large, well stocked, and very helpful fly shop. I think there's one in Gatlinburg proper too but it's a tiny place comparatively.