I've explored and led groups in and around the Greater Yellowstone area for 25 years and have been over all those roads many times. Either route from the Tetons is O.K. On 287/26 from Moran Junction you'll go over Togwotee Pass, but it's not a bad climb or decent, the road has been greatly improved over the past decade or so. You'll travel through beautiful and varied terrain, past the Pinnacles, Ram's Horn along the headwaters of the Wind River. At Dubois you'll suddenly be surrounded by badlands and red rock erosion features. The rest of the way to Casper is through mostly grasslands.
Using 16/20, you'll exit Yellowstone NP along Yellowstone Lake to the East Entrance. Jagged rock formations on the route along the Shoshone River to Buffalo Bill Reservoir and State Park. In Cody is a wonderful group of 5 museums under one roof, The Buffalo Bill Center of the West
, well worth a visit. There are campgrounds in and around Cody, we've stayed at a few. The drive from Cody to US 26 junction is not through any passes, mostly grassland rolling hills. The grades often have a passing lane going up so you don't hold up or get trapped behind slower traffic. Going on to Thermopolis, has hot springs swimming and extinct geysers, but I've not camped there, only visited in auto or van and used a motel. Also, the Wyoming Dinosaur Center
From Thermopolis, US 20 travels Through the Wind River Canyon Boysen State Park and Reservoir
in the Shoshone Indian Reservation.