The Custer area is our choice for campsite. we usually stay inside Custer State Park, but that is a very busy park. they will have some sites that will hold you, and elec, water and dump station.
The Rafter J rv park gets good reviews here, but costs more..but it IS nice.
The bike rally at Sturgis is a big one, so the campgrounds will be fairly busy that one week in August.
You should do the Needles Highway scenic drive and the Peter Norbeck scenic drive, and the visitor center at Custer state park is pretty nice, too.