191 is an easy, pretty much level drive except for a couple miles of maybe 6-7% downgrade just before Hoback Junction. From there turn left and go to Alpine Junction, then route 26 to Idaho Falls, then route 20 to West Yellowstone. Don't be tempted to take the shorter route up to Jackson followed by route 22 over the Teton Pass. 10% grades and many sharp curves on 22.
You could go through Grand Teton and Yellowstone Parks to get to West Yellowstone, but there are several 6% grades plus three crossings of the Continental Divide on that route, plus lots of traffic in Yellowstone. Going around through Idaho is longer, but easier by far.
