Since my son lives in Cody, we've been doing the trip from our NJ house to Yellowstone pretty much every summer.
I have found the best route which minimizes the climbs and the associated bad weather that comes with altitude (hit snow in WY last June...) goes as follows:
1. I-80 all the way to Ogalalla, NE
2. Take Rt 26 up into WY
3. Take I-25 to Casper
4. Get back on Rt. 26 to Shoshoni
5. Turn right onto Rt. 20 to Thermopolis (stop and bath in the hot springs!)
6. Take Rt. 120 into Cody
7. Rt. 14 to the East entrance of Yellowstone
I-80 goes south of Chicago. Just time it so you're going through off peak. And frankly with COVID, I've found less and less people commuting.
Tom and Katharine
'07 Winnebago Tour 40TD, 400hp Cummins
'17 Winnebago View 24V, '02 R-Vision B+
RVing for 20 years & 200,000+ miles