For Mt Rushmore area, DEFINITELY look into Custer State Park near Custer, SD. It is a fabulous park with great facilities for all levels.
The Badlands are more interesting than most expect so I would recommend allowing at least one full day. The Badlands are at most a 2-hr drive from Custer State Park so you could find a CG then explore all day and spend the night. The Badlands Park has typical Nat'l Park level CGs and I would recommend them over anything you can find at Wall, SD, which seems to cater to one-nighters. You will be pretty crowded. Though the town of Wall is a tourist trap, the famous Wall Drugstore might be worth a few hours of your time (it's more like a mall than a drugstore).
If you have enough time, a side trip to the Wounded Knee Memorial is just south of the Badlands.
Some other points-of-interest:
Lewis and Clark Caverns near Cardwell, MT, off I-90.
The Little Big Horn/Custer Battlefield, Crow Agency, MT, also right off I-90.
Ft Laramie National Historic Park, WY (very nicely reconstructed frontier fort southwest of the Black Hills area.)
If you want to head a bit south and take I-80 back to Chicago, there are several interesting sites in Nebraska along the old Oregon trail and the Air and Space Museum near Omaha, site of the old Strategic Air Command (of which I was once a member).
Now there's some stuff to help your planning!