From Shoshoni WY to Thermopolis, you will pass through the Wind River Canyon. Stretching 34 miles between Shoshoni and Thermopolis, follows the Wind River (also known as the Bighorn River) and includes areas where the canyon walls rise dramatically. While the overall drive is scenic and not a continuous steep climb, you should be aware of potential elevation changes and associated hazards when traveling through the Wind River Canyon, especially near the river and rock formations. There are some short, steep downgrades but nothing approaching what you would see on a major mountain road. There are scenic turnouts that you may want to stop to take pictures. (I used to love that drive when we lived in WY.)
Thermopolis is at an elevation of roughly 4,324 feet, while Cody sits at an elevation of 5,010 feet. The route between the two towns is generally ascending and is 84 miles of rolling countryside.
From North Platte I suggest taking US26 to Shoshoni, which will be generally following the Oregon Trail and there are numerous historic sites along the route which you may wish to visit. I recommend at least stopping briefly at Scott's Bluff Natl. Monument, at the Gurnsey Oregon Trail ruts, and at Ft. Laramie. In Douglas there is nice camping in the city park right on the river. There is no mountainous driving between North Platte and Shoshoni.
From Cody to Yellowstone NP there is some mountain travel but none that is particularly challenging.
