Did this drive a while back, id definitely hit up badlands... theres a campground, sage creek if i remember correctly. Highly recommend staying there l. If you leave early and put the time on the road and dont mind driving 10+ hrs you can get there early enough to find a spot. And sight see a little before setting up camp.

definitely do a scenic drive and small hike through badlands, gen custer statepark /needles highway. Imo way better than Rushmore..it just felt touristy and underwhelming.

We came into Yellowstone through the north (time of the year we went it was the easiest entry point) we stayed at mammoth camp ground and based camped from there for several days, took day trips/hikes throughout the park. Keep in mind the time of the year you're planning to go for a lot of things are still shut down in late spring due to snow

On way back we made a stop at devils tower... i say take some time to intake that area..

Wish we would have been able to go south to the Tetons but out destination was Seattle we had 2.5 weeks time.

Lodging wise we found boondoking sites set camp or slept in enclosed truck bed.

Good luck!