Whatever route you take definitely plan on spending some time in Teton National Park which is adjacent to Yellowstone even if it's a little out of your way. They are completely different in looks. For the Teton we love Gros Ventre campground which has 300 sites - no reservations accepted but it's rarely full - perhaps sometimes in the summer.
For Yellowstone, Mammoth campground - no reservations accepted but it's not a large campground. Staying outside the park the previous night is helpful then you can get a site early morning.
You might also try for Madison campground in Yellowstone even without reservations. There are usually spots.
However, are you planning to go this year yet? It's kind of late to be heading out from Illinois. You might want to check for campground closing dates.
Yellowstone Campgrounds
