You realize that Yellowstone is a mountain top?
I25 doesn't cross any mountains - except the Raton Pass on the NM/CO border.
I80 doesn't either, though it crosses the Continental Divide.
The west entrance from Cody, and the NE entrance are the only ones with 'mountain driving'. Everything else is broad valleys (OH, and avoid Teton Pass west from Jackson). OTherwise the driving isn't any more scary than a freeway overpass.
Coming from the east US16 over the Bighorn Mtns is mountain driving, but from the south you don't have to use that.
