We have driven to/through Yellowstone many times. In my opinion, and the route we nearly always drive now, is to continue on I 80 to Rawlins, then US 287 N to the South Yellowstone entrance. That misses most of the mountain grades, and you get to stop at Jackson/ Grand Tetons first. When we leave and head East, that highway is all downhill to Cody, WY.
If you don't mind mountain driving from Cody > West, the route to Thermopolis and the hot springs, I25 to Casper > US 20 to Thermopolis > WY120 takes you to Cody. Then it's all uphill until you reach the lake in Yellowstone.
2000 Winnebago Ultimate Freedom USQ40JD
, ISC 8.3 Cummins 350, Spartan MM Chassis. USA 1SG, retired;PPA,Good Sam Life member."We the people are the rightful masters of both the Congress and the Courts - not to overthrow the Constitution but to overthrow men who pervert theConstitution. "Abraham Lincoln"