Oops, fogot to recommend some campgrounds. We love the views of the Tetons across Jackson Lake, so we stay at Colter Bay. If you don't mind using your RV's generator, suggest the Colter Bay cg, but, if you want hookups, there is an expensive RV park there too.
At Yellowstone, either Canyon Village or Norris cgs are the most centrally located. If you head east from YNP through Cody, we love US 14/14A across the Big Horn Mtns. Great views, interesting geology, and a wonderful Forest Service cg with 30A electric and water called Sibley Lake. We stayed there last September on our way to Yellowstone and would go back in a minute!
The Devil's Tower NM cg would be another good one for a short Class C while on your way from YNP to Wind Cave. And, don't miss Jewel Cave NM while you are in the Black Hills. We thought it was more spectacular than Wind Cave.
Finally, your Class C is ideal for dry camping, so don't be afraid to stop at a Wyoming rest area or Flying J truck stop for a quick overnight. Both are free and many WY rest areas have water taps and RV dumps. Flying Js have RV fuel lanes, RV parking, and even propane, water, RV dumps, and restaurants, if you need them.