Oh my--next to Disney, this trip you're planning is about as good as it gets!
We've done this trip (minus Grand Tetons and add Glacier National Park) several times--you're going to love it!
We drive west from Illinois along 94 and each stop is better than the last.
Our stops include (but not limited to and not necessarily in geographic order):
1. Al's Oasis near Chamberlain, S.D. (Gotta have a picture on the big buffalo)
2. Mitchell Corn Palace
3. Wall Drug
4. 1880's Town
5. Badlands
6. Custer State Park
7. Wind Cave
8. Needles Highway
9. 4 Heads (oops, I mean Rushmore)
10. Crazyhorse--eat at the diner there--sample their Navajo tacos. In fact, sample Navajo tacos wherever you see them offered.
11. Devil's Tower
12. Sod House outside of Yellowstone
13. As many trails as possible while in Yellowstone. Get out of your car and walk. But don't get too close to the wildlife. You won't believe the stupid tourist stories the Rangers can tell of people being injured by the wildlife. Oh, and do at least one Ranger talk/walk. They're great fun.
14. Huckleberry ice cream. Huckleberry pancakes. Huckleberry candies.
15. National Bison Range between Missoula and Glacier.
16. As many trails as possible in Glacier. Especially the High Line, Avalanche Lake, and the one at Logan Pass. Okay, I know you're not planning to go to Glacier. But you should consider it. It's amazing.
17. Horseback riding outside or inside any National Park. But avoid the mules that have gas. (Long story.)
18. Whitewater rafting. Or not-so-whitewater rafting. Relaxing and breathtaking.
As for how long--it depends. You can see Mt. Rushmore (stop, get out of car, look at heads, take picture, get back in car) in about an hour. Or you could spend a day there. Heck, we can kill a half a day at Wall Drug alone! But definitely plan at least 3 days at Yellowstone--traffic can back up and slow down travel, and that's one park you really want to get off the road and explore on foot.
Have a great trip! I envy you!