Welcome! Since you mentioned Mt. Rushmore, you'll probably want to take I-90 across SD, which has the Badlands and Black Hills, and Devils Tower in n.e. Wyoming. Stay on I-90 up into Montana to visit Little Bighorn National Battlefield, which is quite an interesting stop. Devils Tower has a KOA right next to it with a dandy view of the Tower; worth an overnight, IMO. For the Black Hills (i.e. Mt. Rushmore), there are a lot of nice CGs. Our favorites there are Custer SP (state park) and Rafter J Bar Ranch CG. For Little Bighorn, there are 2 decent RV parks nearby; we chose the KOA in Hardin and were pleased with it. It's worth overnighting near this park, if you are interested in the history, as the ranger talks at the battlefield are wonderful, and each one is a bit different. There are self-guided or guided tours available there too, which add to your understanding of what happened to Custer & his men, and why.
Agreed on making reservations early for Yellowstone. Check the park's website (nps.gov/yellowstone) for their "plan your visit" information for details, but basically there are 2 kinds of camping options within the park: campgrounds without hookups, and Fishing Bridge RV Park, which has full hookups. Sites are much tighter in the latter, but it's pretty centrally located, and the hookups are great to have, whether you need the heat or the a/c on. (Late August/early September, nights will be cool, likely around freezing. Daytime temps still pleasant, though.) Be aware that some of the CGs in the park and some of the hotels/shops/restaurants will close by mid-September, though not all. I believe Fishing Bridge closes mid-September. You can make reservations now for next summer for lodging - including those CGs that take reservations -inside the park. If you know your travel dates, I'd advise doing so ASAP, even though will be traveling after peak season for YNP. Attendance at the park diminishes by late August but doesn't drop off entirely until late September, when most facilities there are no longer open.
Grizzly is our favorite outside-the-park CG, and we have stayed there many times. Very nice, neat RV park. If you are not pulling a "toad" (towed car), staying in West Yellowstone could be quite convenient, as you can rent a car there. MUCH better for touring the park in than a motorhome!!
Definitely, absolutely a MUST: allow yourselves a minimum of 3 full days in Yellowstone; 4 or 5 is better. The park is huge and takes quite a while to drive around even half of it. Part of what slows you down is the 45 mph speed limit and the distances in the park, but mostly it's photo stops, animals by/on the road, other tourists' driving habits, etc. :) Definitely plan to visit Grand Teton National Park while there, since it's right next door (5 miles from the southern border of YNP) and can be visited in a day from West Yellowstone - or you could camp there for a couple nights before/after your YNP stay. My recommendation for GTNP is to camp at Colter Bay RV Park if you need hookups; otherwise Colter Bay (NPS) CG or Gros Ventre CG come highly recommended (no hookups) by a lot of folks on the forum.
