We were in SD earlier this month for the first time and will definitely return. So much to see and do but our trip cut short by Hurricane Matthew.
If you don't mind dry camping, consider a day or two at Sage Creek in Badlands National Park. It's about twelve miles down a graded dirt road but there were a couple of class A's there when we visited. Fairly rustic with no specific sites, only a few picnic tables and two dry toilets but its free and the experience is awesome. Free range Buffalo wandering through the campground, prairie dog towns, big horn sheep and incredible night skies.
I don't know how busy Horsethief Lake campground in Black Hills National Forest gets in Summer but we only had two other campers during our stay and I'd definitely recommend it. It's only a few miles from Mount Rushmore.
Lee's Ferry in Glenn Canyon National Recreation Area is another developed campground we liked. It doesn't take reservations so you should have a good chance of getting in if you arrive mid-week. Lone Rock Beach primitive camping area is another possibility.
Unlike here in the east, there is a tremendous amount of public lands in the west and its fairly easy to find a place to camp if you boondock or at least dry camp. Kaibab National Forest on the North Rim of the Grand Canyon has several campgrounds along with almost unlimited dispersed camping spots.
Hope you enjoy your trip.