If you are comfortable with mountain driving Hw 14 is doable. Stay away from Hwy 14A although you will miss some wonderful country.
Better yet for you flatlanders go to Buffalo which in itself is worth a visit. You could stay at a nice campground there called Indian Campground
Indiancampground. take a swim and visit the museum in town, also the old hotel,
Occidental Hotel and proceed the next day via Hwy 16. Hwy 16 has several USFS campgrounds on your way to Cody. There are also some great boondocking sites along the way. Tensleep is interesting and Cody is worth a day or so. Stay and Ponderosa Campground
Ponderosa Campground, Make reservations and don't miss the
Buffalo Bill.
This is great country and you will know you have visited the wild west when you see the Mississippi River again.
As a side note I went to school in Fulton, Illinois a century ago!