Near the south entrance the best campgrounds are Fishing Bridge for full hookups or Madison or Bridge Bay.
Definitely spend some time in Teton Nat'l Park just south of Yellowstone and for that park your choices are Colter Bay RV (concessionier, full hookups) and right next door is Colter Bay campground (no hookups - 300 wooded sites) and a little further south toward the town of Jackson is Gros Ventre campground (no hookups - 300 sites) in the park.
Trying to cover both of the above parks from one campground will be a LOT of unnecessary driving and very long days. It's more pleasant to stay in each park.