Good route suggestion by Busskipper. From Alpine you could continue on route 26 to Idaho Falls then route 20 up to West Yellowstone. No mountain driving that way. That said, my preferred route would be to go north on route 89 through the Snake River Canyon (easy grade) to Jackson and Grand Teton NP and spend a few days with those beautiful mountains then continue north to the south entrance to Yellowstone.
X2 on do not use route 22 over the Teton pass west of Jackson. Been there, done that, never again.
