We spent a week in South Dakota this summer and didn't even hit the Badlands. There is so much to do in SD! Heading across I-90--
There is a lot to do in Sioux Falls. Mitchell has the world's only Corn Palace. Chamberlain has a really nice Native American cultural museum and you must stop at Al's Oasis for buffalo burgers and hot apple pie with cinnamon ice cream! Near Kadoka there is an 1880s town. Then you hit the Badlands.
There is a little gift shop/informational stop where you can feed the prairie dogs. In the Badlands there isn't much to do except drive the scenic route and hike. Hiking is not easy. A couple of years ago a group of people I know were hiking out there. On one hike they had someone fall and break a wrist and another fall and break an ankle. And it's a long way to the nearest hospital. You also have to watch for rattlesnakes. If you go down by the Rosebud and Pine Ridge Indian Reservations you can see the Wounded Knee Massacre Memorial.
Just on the other side of the Badlands is Wall Drug. Easily the most famous tacky tourist stop anywhere. Then you hit Rapid City and the Black Hills. There are tons of things to do in that area. We were there four days and didn't even scratch the surface of all there is to do. Mt. Rushmore is a must stop and so is the Crazy Horse Memorial. We had friends who spent a week just in Custer State Park camping.
After you leave Rapid, it will take a good day and a half to get to Yellowstone. Just inside Wyoming is Devil's Tower which is an amazing site! There are two routes over the Big Horns--Hwy 16 through Buffalo and into Worland, or Hwy 14 through Sheridan and into Greybull. I'm partial to the Hwy 14 route because I grew up on that side of the mountain. But, either route is good. No matter which route you take make sure you have a full tank of gas before you head up the mountain. There are limited places to get gas on the mountain and it tends to be quite a bit more expensive because of the limited supply.
Some people think the Hwy 16 route is an easier drive, but I don't know if it is or isn't. When you come into Worland you can take the southern route and go into Yellowstone through the south entrance (closer to Jackson Hole and Teton National Park), but it is a much longer route. You can also take the northern route and end up going through Greybull and Cody.
Highway 14 is my favorite route. (DO NOT take Hwy 14A to Lovell unless you are VERY comfortable driving mountain roads!!!) Hwy 14 will take you through Shell Canyon. There is a very nice place to stop at Shell Falls with easy hiking trails. At the bottom of the mountain you can stop at Red Gulch and see dinosaur tracks. In Greybull there is a nice little dinosaur center and they can arrange tours of the Red Gulch site.
Plan to spend some time in Cody. The Buffalo Bill Museum is really nice. It is also well worth it to take a little side trip and see the Heart Mountain Relocation Center Museum between Cody and Powell. They have an amazing museum there.
I hope you have at least two weeks for this trip. Without any stops at all it will take you at least a day to get across South Dakota and into Wyoming. As a college student driving by myself from Sioux Falls to Greybull was a 12 hour drive straight through. It's another 2 hours just to the entrance into Yellowstone. Once you get into Yellowstone it is at least an hour before you get to the first tourist stop (restaurant/gift shop/etc.). Driving times in Yellowstone greatly depend on how many buffalo herds and tourist are blocking the road and how many people with gigantic RVs can't maneuver through. (Personally I think it would be a nightmare to drive a huge RV over the Big Horns and through Yellowstone. But, people do it all the time.)
Plan way ahead!! It is not easy to find places to stay in Yellowstone unless you are camping and even that is limited. If you want to stay in one of the hotels in the Park next summer book it NOW. If you are driving across South Dakota in August make reservations sooner rather than later. Speaking from experience, once the Sturgis Motorcycle Rally starts it can be impossible to get hotel rooms anywhere along the I-90 corridor all the way into Yellowstone.
I grew up in Wyoming near Yellowstone and lived in South Dakota for 11 years. I've driven that route more times than I can count. Private message me if you want more details.