IMO the most level route from I80 would be to take route 191 from Rock Springs to Hoback Junction, then south to Alpine Junction, route 26 to Idaho Falls and route 20 on up to West Yellowstone.
The above route misses Grand Teton NP, which is a shame. If you have time for Grand Teton, just turn north at Hoback Junction to Jackson. After Grand Teton, you could go back south through Idaho Falls as above, or go north through the Yellowstone South entrance and through the Park to West Yellowstone. There are ups and downs inside Yellowstone, but no serious mountain driving.
Route 287 from Rawlins is also a good route that will take you over the Togwotee Pass between Dubois and Moran Junction. It's a long gradual climb and a gradual descent into Jackson Hole.
Route 191 north from West Yellowstone to I90 is the easiest drive to start on the way to the Black Hills in SD.
Old Guy is correct that the east entrance to Cody is a steep and curvy down grade. The easiest way I can think of to get to Cody would be to take I90 to Laurel MT then route 212, 72 and 120 south the Cody. Quite a bit of extra driving to avoid the grade on the east entrance. A long day trip through the Park to Cody and back in the Tundra is a consideration, although that's a lot of miles too.
You don't want to take the northeast entrance leaving the Park, since that goes over either the Beartooth Highway or the Chief Joseph Highway. Absolutely beautiful drives, but both definitely mountain driving.
