While the suggestions posted are good, I'd take a different route completely -- I-90 across South Dakota to I-25 at Sioux Falls, then south to I-680/I-80.
Hwy 14 over the Bighorns at Greybull or Hwy 16 over them at Worland would both work, but 16 is a little easier and just as scenic, imho. This would lead you to the historic town of Buffalo (Johnson County Range Wars), where you'd pick up I-90.
From there I'd suggest a slight excursion at Moorcroft to visit the nation's first National Monument, Devils Tower, then head to Sundance (namesake of the Sundance Kid) and I-90 again. This will place you at the edge of the Black Hills, where you'd have any number of interesting sites to explore including Mt Rushmore, Crazy Horse Monument and Custer State Park, to mention just a few.
Once leaving the Black Hills you'd be only an hour from the South Dakota Badlands, another great short excursion from the interstate (not to mention Wall Drug).
The rest area just past the Missouri River bridge at Chamberlain provides a very nice scenic view of the river and the rugged plains beyond it. East of the river is farm country; west of it is cowboy country, and it's so different that residents of the state identify as either "east river" or "west river".
Normally I'd recommend a quick stop at the Mitchell Corn Palace, but it's currently being renovated and has lost its charm for the summer. Mitchell does have a nice Cabellas, if you're so inclined, as does Rapid City.
If you take I-25 south at Sioux Falls, don't miss the I-680 exchange and stay on I-25. I-680 is a direct route to I-80 through some pretty country and will save you 40-50 miles (wild guess).
Try to stop at the first rest area when arriving in each state to gather tourist information. It'll make your trip through each state much more interesting and you'll be less apt to miss something that would appeal to you.