In Yellowstone and Teton all you have to do is hike a little ways away from the easy access points and you'll have prime fishing all to yourself.
In Jackson, most guided trips float the snake. Loads of rivers and streams with tons of access to great trout fishing if you're willing to hike a bit.
If you want big trophy rainbows, most people were hitting the green river closer to Pinedale back when I lived around there. It's been a few years though, so the pressure may have pushed them further down river.
If you're really serious about big fish, try and time it with the full moon and fish at night for big brown trout with streamers.
Gear-
-waders without felt bottomed shoes
-5, 6, or 7 wt rod(s) with floating line for dry flies and something a little heavier for streamers
-short rod in 4 or 5 wt for some beaver pond and brush fishing....practice a slingshot cast
-polarized sunglasses
-backpack to carry lunch and fishing gear
-bear spray
There's loads of great fly fishing all over the area you mentioned. You just have to figure out where you want to focus. Make sure to stop in every fly shop and check out the board to see what's working. The fly shop on the square in Jackson always seemed to have the most knowledgeable guides that were also helpful, never used them for anything other than flies or info though since I had my own boat.
My go to's
-mosquito fly size 18-22 for beaver ponds in the evenings
-mayflies.....if they're hatching
-wooly boogers - mostly olive and black, but the cowboy Joe works too (brown & yellow)
-tan deer fly
-royal coachman
-elk hair caddis
And the holographic panther Martin for backup