Welcome to Travelblog! Roadtrips are always an adventure. The United States is so large and varied that there is no single "best" sights. What to see depends entirely on your interests. Reading blogs from various places will reveal many interesting spots.
Alabama to Yellowstone is nearly 2000 miles one way. Depending on how many miles you want to do each day, it could easily take four days just to get there. Yellowstone requires a minimum of three days to see fully, and Grand Teton to the south is worth another day. That leaves four days for everything else. I would spread them out to break up the driving, leading to something like the following:
1. First drive to St. Louis. The Arch is unmissable.
2. Head northwest to eastern South Dakota. The Corn Palace in Mitchell is popular.
3,4. Drive west to the Black Hills, visiting Wall Drug, Mount Rushmore, and whatever else time allows.
5. Continue west through Northern Wyoming. The route leaves the interstate to cross the Bighorn Mountains. Make sure to drive this in daylight because the scenery is spectacular (and the road is a little scary at night).
6. Drive the Buffalo Bill Scenic Highway to Yellowstone and enter the park
7,8. Tour the park. The park service runs multiple scenic bus tours, check the website.
9. Drive south through Grand Teton.
10, 11. Drive southeast on state roads in Wyoming until you reach Interstate 80. Take this east to Cheyenne, and then turn south for Denver.
12, 13,14 Drive south until New Mexico. Once across the border, head southeast for Amarillo, and then Dallas.
15,16 Head for home, Driving northwest from Dallas will pass through Little Rock and Memphis, and won't add much extra time.
Looking forward to the blogs!