I assume you meant "across on I-90 to Badlands", correct? Because that would be the normal route.
Unless you're really frightened of mountain driving, I'd also recommend Hwy 16 from Buffalo to Worland. Much of that is mountainous, but it's all a good, wide 4-season highway used by vehicles of all kinds. Take it easy, use lower gears descending and DO NOT RIDE your brakes, and you should be fine.
As Mr Shoe noted, there will be at least a couple pull-offs where you're encouraged to stop and check for overheated brakes. Do it. Use that time for a little break, take some photos, maybe have a sandwich. I've hauled my campers up and down and over both Hwy 14 and 16 dozens of times. In my opinion 16 is the easier route. Scratch that. EASIEST route. Just use some common sense.
I also encourage you to spend some time in the Black Hills -- Rushmore, Crazy Horse Monument, Custer State Park, Jewel Cave and Devils Tower, just to mention a few. Also plan on a day in Cody for the Buffalo Bill Historical Center, etc.