We spent five nights at Custer State Park's Game Lodge campground last August and enjoyed the Black Hills area area very much. Reserve early. Electric at each site, and a water hookup by the laundry building for filling your tank, and a dump station. There are several other campgrounds as well.
Plenty to do! The state park itself has so much to see including the park's big herd of bison that spent part of each day beside the campground - great photo opportunities up very close. Beautiful Sylvan Lake; the Wildlife Loop with prairie dogs, pronghorn antelope and more; the Needles rock formations, the tunnels. Each night, the rangers had programs or movies at the campground theater building.
The historic Game Lodge itself is worth stopping in for a drink or dinner. Take a look at the stone fireplaces in the bar and lobby area, with crystals from nearby Jewel Cave.
As mentioned, many other interesting things to see within an easy drive, like the Caves. I got a big kick out of spending a morning at Rushmore, hiking around the base for different views. The visitor facilities there are all very updated.
You can drive west about 90 minutes to see Badlands NP and the Minuteman Missile National Historic Site but it's also fun to stay overnight to see the colors change on the Badlands at sunrise and sunset. The campground there has some electric sites and now takes reservations.