You mention Jackson and if staying there going through Yellowstone & Grand Teton is OK but it may be slow because of crowds. Another route would be West Yellowstone via Hwy 20 to Idaho Falls then 26 to Alpine and north to Jackson. There's a little shortcut from 20 to 26 via Victor but DON'T get on 22 between Victor and Jackson as a short cut - you won't like the steep Teton Pass. Instead, once in Victor take 31 to 26 to Alpine. If staying in Grand Teton then going through Yellowstone makes more sense.
387 to 50 into Gillette would also work.
The other highways are good.
