Route 20 is an easy drive to West Yellowstone. Haven't done route 191 for a number of years, but it follows the Madison River valley and I don't recall anything exciting.
You didn't mention how you plan to get from Livingston to Cody, but the easiest route is I90 to Laurel MT, then routes 212, 310, 72 and 120 to Cody. The only other route is through Yellowstone, in the North Entrance and out the East Entrance. The North Entrance has a one mile 7% climb to Mammoth Hot Springs, then another mile or so of 6-7% to the plateau. You need to go around the west side of the upper loop, as the east side has a steep climb and descent over Dunraven Pass. The East Entrance has a 6-7% descent for several miles between Sylvan pass and the Entrance station, followed by a more shallow descent pretty much all the way to Cody.
We like the Ponderosa RV Park in Cody. Only a couple of blocks to the Buffalo Bill Center.
2003 Winnebago Minnie 24F - Ford E-450๐