Camping at the Sage Creek campsite in the Badlands National Park is free but you do have to pay to get into the Badlands if you don't have an annual pass. Although if you're planning on checking out the Badlands on your trip you're going to have to pay that anyway. The Sage Creek campsite is located at the western end of the north unit of the park. There's another campsite near the visitor center but that one is $20-$30 a night. The Sage Creek site has vault toilets and that's about it. No water and no fires allowed. Also it has bison, but sadly they never came to pay me a visit.
The U.S. Forrest Service allows free "dispersed" camping in any, non developed areas of the Black Hills National Forrest. There is a limit of two weeks and you are not allowed to camp near any recreation areas or their paid campsites. There's quite a bit of private land scattered through the national forest as well. It's like a patchwork quilt. Best bet is to stop by a ranger's station to get a map and ask someone there if you're not sure. There's a station for the northern Black Hills in Spearfish, one for the southern hills in Custer, and one in Sundance Wyoming for the little part of the Black Hills in that state as well as the Bear Lodge Mountains. Also while dispersed camping is allowed in the Black Hills National Forrest, campfires are only allowed on the Wyoming side. Unless South Dakota has lifted its burn ban.
The U.S. Forrest Service also operates the Buffalo Gap National Grassland which covers quite a big chunk of south western South Dakota. I know there is at least one free campsite at French Creek which is pretty far from anywhere. There were vault toilets and fire rings there, but no water. I'm not sure what their rules are for dispersed camping outside of French Creek campsite, but if you can camp outside the campground you probably will be under the the burn ban.
Free, dispersed camping is also allowed on Bureau of Land Management land and there's a little sliver of BLM land on the northern edge of the Black Hills just east of Sturgis (right across the highway from the national cemetery). They also have a small paid campground there with water, toilets and fire grates which was only $6 a night. The site is actually free in the off season but that ends in May, so too late for you. There is additional BLM land scattered across the plains of western South Dakota and Wyoming but it might be a bit out of your way.