If you don't have Grand Teton Park on your list, you should take I 80 to Rock Springs WY, then 191 up to 89 and north to Yellowstone. You'll pass through Jackson and GT park.
About 20 miles northwest of Pinedale WY there is a cheap little Gov. campground with water and a dump station right on the Green River. Nice stop before going on to Jackson. North and east of Jackson is Gros Ventre campground, a park campground with water and dump and lots of shade. About 65 miles from Yellowstone.
Backing up, there is a good rv park at Cabela's with full hookups in Sidney NE..or you can park at Walmart.
Between Cheyenne and Laramie there is a rest stop called the Lincoln Memorial and it is a very NICE rest stop, nice and cool.