If you're new at driving in the western mountains, consider getting online the "Mountain Directory for Truckers & RVers" Western Edition. It publishes Eastern, also.
Hill City to Fishing Bridge in Yellowstone:
16 to Newcastle to I-90 at Gillette to Buffalo
16 west to Worland; 16/20 to Greybull; 14/16/20 to Cody and continue straight to Fishing Bridge. You'll have a couple somewhat steep areas but very doable with an RV and many do it. You'll have the Tensleep area on 16 between Buffalo and Worland and then leaving Cody into Fishing Bridge is a descent. We've done them a few times and they're short areas. Just take it slow downhill and use your gears to slow yourself at the top of the hill, not in the midst of driving down.
Hill City to Teton Nat'l Park - East Entrance:
385/16 to 85 to Lusk, Wyoming
West on 18/20 to I-25 at Douglas
220 at Casper and 287 into the east entrance. No biggies.
Hill City to Jackson from the south:
Just use the above but weave yourself southwest to get to 191/89 into Jackson. Nothing to be concerned about.
You could just get go to the Tetons first and leave your volunteer gig from Fishing Bridge to do Cody, etc. but I think the climbs are more noticeable from west to east. It would probably be a nice trip if you just take the above route to Fishing Bridge and then continue south out of Yellowstone into the Tetons after your gig is up. The Tetons are so much different than Yellowstone. It's worth doing both.
Also, you're sure to have days off and from Fishing Bridge to Jackson is an easy day trip if you want to do major grocery shopping, filling with fuel, etc. From Fishing Bridge, Jackson is the cheapest place to purchase groceries. Just south of town on the right is a great Smiths grocery with a reasonably-priced fuel station and a liquor store.
Have a great summer! Wish we were going there once again.
Full-timed for 16 Years
. . . Back in S&B Again
Traveled in a 2004 Newmar Dutch Star 40' Diesel