Pretty much anywhere in Black Hills National Forest is open for camping, although in the SD portions of it, fires are not allowed unless you are in an established campground according to the forest service website. There are a few places along the trail up to Harney Peak where you can drop off of the trail and make your way into the middle of nowhere to find a camping spot. Whether or not you camp there, the Harney Peak hike is one I definitely recommend, plan on it taking at least 4 hours to get to the top and back. While you're up there, the Needles Hwy is a cool drive and will take you down into Custer State Park if you follow it the whole way.
Spearfish Canyon is a nice drive, especially if you come in late summer or early fall. Roughlock and Bridal Veil falls are cool to stop and see when you're driving through the canyon. The missle silo by Wall is a cool tour, and the museum at Ellsworth AFB is worth a stop too. On Hwy 85 south of Lead and Deadwood there is a road that will take you to the top of Terry Peak, which is kinda neat too. You can see the huge Homestake Gold Mine in Lead from up there. Deadwood is a cool place to stop and wander around, theres plenty of casinos if you feel like trying your luck, and lots of cool historical stuff to see.
I'm sure I'll think of more, but for the time being here's a few pictures to enjoy...
From the top of Harney Peak..
Cathedral Spires along the Needles Hwy.
There are definitely Mt. Lions in the area, and they aren't as afraid of people as some would think. These prints were fresh, and right next to the hiking trail that is busy all day long....
Sheridan Lake