One place not to miss: The Minuteman Missile National Historic Site. It's near the badlands/wall drug area, but absolutely worth going. Word of caution: very very hard to get tour tickets (arrive before they open - tours are 6 people at a time and only so many per day!) and parking a rig at the launch facility will be a challenge - possible but a challenge. Toad suggested! Parking at the visitor center not so bad. (You get your tour tickets and time at visitor center and they tell you how to get to launch facility and you drive threre and meet your tour guide - usually retired Missileers!)
NPS site: Minuteman Missile National Historic Site (U.S. National Park Service)
From our trip earlier this year (first chunk of this post): https://breighton.qseg.org/?p=21250
And if you have a toad, make sure to drive the Needles/Iron Mountain/Norbec/us-16a around Mt. Rushmore. If you have a large truck make sure you take any mirror extenders off before you go or you might loose them in the one way tunnels. (Kinda kidding but just barely - like by an inch or two max!). One of the tunnels is 8' 4" wide. I know my truck towing mirrors stick way out close to that (as they are allowed to be wider than body size limit of 8'6" in most states and on federal highways)
If you can get in, custer state park game lodge campground makes a fantastic base camp. Spent a week there and it's close to everything. No water hookups (electric only) but have nice bathhouses and fresh water to fill your tanks as you arrive and dump station you can use when you leave.