We have done the 16 route over the Big Horns to Cody and it is a beautiful trip coming over. Little steep in a couple of areas, especially where the switchbacks start going down into the canyon.
We have also run I-90 to Livingston and taken 89 into the park through Gardiner. Again curvy and a bit steep as you come into the park.
I-90 to Bozeman and down 191 into the Park at W Yellowstone is our first route ever there. We still prefer that route if we a just touring around that part of the country. Seemed to have seen more animals there (wife loves wild life) coming into the Park this way.