The respective park pages should tell you all you need. Except for the Black Hills themselves, South Dakota is flat prairie, and all state and federal numbered highways should be just fine.
https://www.nps.gov/badl/planyourvisit/directions.htmThe Badlands is an eroded river valley(s), and all sightseeing is from overlooks along the rim. The parks page describes one paved rim road, and a couple of gravel alternatives. You can also enquire at the the visitor center.
https://gfp.sd.gov/csp-scenic-drives/Custer has some scenic drives with tunnels - those dimensions are given on this park page.
https://gfp.sd.gov/userdocs/docs/custer-map.pdfThe park map shows tunnels in the surrounding area, including a few on the drive toward Rushmore.
Parks also have visitor guides (often large format newspaper like ones) with maps and road information. These can also be found online.
https://gfp.sd.gov/userdocs/docs/tatanka_2021.pdf