There are lots of threads about driving from the east to the Black Hills and on to Yellowstone. Also threads about driving from I80 to Grand Tetons.
In Wyoming the main highways to avoid in an RV, are 22 Teton Pass into Jackson, and US14 (and 14A) over the Big Horn Mtns. US16 is regarded as a reasonable way over the Big Horns. All routes south of that skirt those mountains.
On Google Maps it is easy to identify where the mountains are. Even zoomed out they are shaded as National Forests. Turn on terrain mode to see them in more detail. Zoom in even more to see contour lines. Streetview gives you road level images of the highways.
