I agree with others. Going to interesting places in the Rocky Mountain West without a car will significantly reduce your chances of seeing the best stuff.
How I would go and what I would see? I suggest you take the route on this MAP
To see along the way. If you've never been to a large hunting and fishing store you might want to spend an hour or two at Cabela's in Sidney, NE. They have a large parking lot and interesting things to see.
Visit the Mammoth Site in Hot Springs, SD. This is an active archeological dig. I find it really interesting. Imagine the Black hills without many trees and huge hairy elephants wandering around!
Reserve an RV site in or near Custer, SD. From there using your CAR tour the Black Hills, see Mount Rushmore, Custer State Park, the Needles Highway, Crazy Horse and maybe ride the narrow gauge train at Hill City. You might want to tour a cave such as Jewel Cave or Wind Cave.
Make your way to Devil's Tower, WY. There is a KOA at the Devil's Tower entrance. Devil's Tower is interesting!
From Devil's tower continue on to Worland, WY. Between Buffalo and Ten Sleep is Powder River Pass. It rises 5000' above Buffalo before descending 5200' to Ten Sleep. That sounds like a lot and it is. However the highway is good, the grades are reasonable and the curves easily navigable. I do it regularly in a 34' gas motorhome towing a Jeep.
On the map I suggested you go to Thermopolis, WY. You might want to spend the night there. Thermopolis features hot springs pools that are very relaxing and fun for kids if you have 'em. It also has a very nice dinosaur museum. From Thermopolis take WY120 to Cody. If you have no interest in Thermopolis, from Worland go to Greybull, WY and take US14 to Cody.
Plan an overnight stay in Cody and visit the Buffalo Bill Center of the West. It is an extensive complex that houses 5 museums. Some people can spend several days just doing the Buffalo Bill Center of the West. For me, a few hours does it.
Then it is a straight shot from Cody to the East Entrance of Yellowstone. I have the map ending at the Fishing Bridge RV park. If you can't get reservations there I recommend you continue on through the park via Canyon village using the Grand Loop Road to West Yellowstone where there are several RV parks just outside the West Entrance to Yellowstone.
Tour Yellowstone using your CAR!
Wave to my cousins by the dozens as your drive past Gillette, WY!