We have made Custer-Mt. Rushmore area an almost annual trip for past 16 years. Sometimes camping, sometimes hotels. Custer is our landing zone of choice.
In picking routes, please note I70 east and westbound between Gypsum and Glenwood Springs, CO, is currently heavily damaged, and from photos, complete repairs will be quite some time in coming. Whether I70 or I80, snow can appear well into May. If you take southern route to I25 through Colorado to Wyoming to SD, there's plenty to see along the way. Devil's Tower in WY is enjoyable, worth an overnight. We've stayed at KOA there in the past and it was fine.
In Mt Rushmore area there are also Jewel and Wind Caves (national park service operations) to visit.
We used to use Flintstones campground with the grandkids, but I understand it closed permanently last fall. I think we stayed at Custer KOA some years ago.
Any time in the summer, plan to dress in layers. It can get quite chilly in the evenings. And rain gear. There is the Mickelson Trail for hiking, cycling, as well.
I will say Custer grocery shopping is sort of limited, so if you have any "must have" items plan accordingly.