I agree with the above, but I would add if you are worried about mountain driving, then it also depends what route(s) you take to get there and where you drive while in the park.
I've been to Yellowstone a few times, and much of the main route is actually pretty flat if you stay on the main road and come into the South entrance, but there are a few places with some switchbacks and incline/declines. If you start in Jackson and head into the park from there, you will initially be driving on all prairie, as you go deeper into the park, you will cross some hills or non-steep inclines, but nothing major that I recall. If you head into or out the Northern / Desert entrance of the park, there is definitely some more mountainous driving with more switchbacks, but it doesn't last very long (maybe 5 miles of switchbacks). I haven't driven every route though. If you've ever been to Portland, it's kind of similar to those inclines/declines between the coast and Portland, but it's nothing like those crazy 10,000+ feet drives in Colorado. None of the drives there are even as bad as the Mt. Washington drive in NH, that one has more incline / switchbacks than most of Yellowstone. Much of the Yellowstone roads are in the prairie or forest and the inclines are slow and it simply more like driving in hills than mountains.
I find much of Montana and Wyoming the same way, most of the main roads are pretty flat even though you are surrounded by mountains, though there are exceptions. It might be a tough drive getting to Jackson from your route, but I came from Salt Lake City, so don't know that route as well. From the SLC route, I recall one maybe two mtn passes that I crossed. Once you get near the park itself, the drive is mostly tame.
It's not nearly as bad as somewhere like Arizona or Colorado, even New Mexico has more switchbacks on some of their semi-major roads.