If you have a little bit of time, take Hwy 44 across Dakota. It is south of I90 and more or less runs parallel. It is very scenic, with little traffic.
From Sioux Falls go south on I29 a few miles and exit west at 276th Street. Go west and you’ll end up on Hwy 44. Or access Hwy 44 by dropping south from Mitchell, SD. Either way will get you to Hwy 44.
44 takes you to “Interior” SD at the eastern edge of the Badlands. South and west of Interior you can find BIA roads, which are dirt/gravel. Make sure to have a good spare as these roads are remote with little traffic.
From Hill City, SD (Black Hills) you can take Deerfield Road/West Deerfield Road across to WY. These are Forest Service roads. Also very scenic. Once in Wyoming take a side trip to Devil‘s Tower.
Forest Service maps are available which will show the roads in the Black Hills. I think that the BIA roads in the reservation are listed in the Gazeeter maps. The roads also show up on the iPhone map app, and probably on Android.
Enjoy the trip!