We drove from Auburn AL towing our car several years ago, leaving home in early June. Our route took us to Nashville, St. Louis, Denver, Cheyenne, WY then west on I-80 to Rock Springs, WY. From there we went NW to Jackson Hole, WY. From here you can tour the Tetons, Then go a bit further north into Yellowstone. If you go this route there is no need to disconnect your toad until you are in a campground.
While at Yellowstone, we stayed at Fishing Village Campground, a full service RV park. As I recall there are a number of US Forest Service campgrounds too. Most don't have electricity, but probably have a source for drinking water.
If you go in late June or even July, you can experience some very cool nights. We were in Fishing Bridge on July 4, and there was a light frost that morning!!! We left the motorhome at the CG and toured around all of Yellowstone for several days.