I used to work at Custer State Park as a ranger, and I would recommend either Center Lake campground for seclusion and swimming, Bluebell campground for scenery and seclusion, or Game Lodge campground for easy access to other areas in the Black Hills. I would recommend to stay clear of Sylvan Lake campground--too crowded and always noisy. Nice for a few hours at the lake but that's about it.
FOr a great day trip if you stay at Bluebell: Wake up early and take the Wildlife Loop road which is one mile south of the campground on Highway 87 and is your best opportunity to see buffalo, antelope and other wildlife. It's a 16 mile road, but expect to take up to an hour on it. If you have the proper vehicle, take some of th CSP roads within the Loop road area. They are dirt roads that are marked and your chance of seeing more wildlife is better here. Get a map at the entrance booth.
After you exit the Loop road on the east side (by Game Lodge Campground) take Highway 16A east to the entrance booth and turn north on the Iron Mountain Road and head up to Mt. Rushmore. A beautiful and winding drive and you will be able to see Mt. Rushmore thought the 3 tunnels that are on the road. Beautiful.
Have a great time no matter what you do or where you stay.