Have you mapped this to see how long the drive is? If you drove 500 miles a day it would take over 4 days to get there - time to turn around and come back home. 38 hours of driving just to get there. If you drove 10 hours a day = 4 days there and then you head back home. That's a miserable road trip. Check into flights and get closer so you can enjoy the trip more - or plan more than 10 days for the trip.

My first thought was "that's a really long drive for 10 days" then I saw starrs response with the distances just confirmed my initial impression.

You are way better off to fly to Denver or somewhere close and rent a car. Otherwise 10 days is just not enough time to make the trip realistically.

Not to mention the amount of traveling you'll do in Yellowstone. Its a huge park, and there isn't a way to cross it easily. You'll do a lot of driving inside the park to see the various attractions as they are located all over the park. Frankly I like the Tetons just as much, if not better. Stayed for 7 days camping in Gros Ventre, going again this Septemer. But either way your driving isn't over once you arrive, so if you're up to driving, have fun. Or, fly to Salt Lake City, rent an RV and really enjoy the park(s).

PS - Grand Teton webside has some great webcams for you to check out the areas. So does Yellowstone, but they tend to go down. And be prepared for weather changes of all kinds. They've had snow in every month of the year!

Once you get past Chicago you can make pretty good time, I think the speed limit is up around 70 or 75 mph on I-90. So doing 700 miles in a day is not hard.

Plan on at least a few hours to go through Badlands NP, before you head down to Rushmore. Rushmore itself should not take more than a few hours to visit.

The trickiest part will be finding lodging in/near Yellowstone, call every day and hope for cancellations.

We did a similar trip a few years ago from CT and I'd spend that much time in a car to go to Yellowstone again in a heartbeat.

We drove south from Yellowstone through Grand Teton and continued southeast to I-80. 80 was much busier with more trucks than 90.

Yellowstone is about a 10 hour trip from Denver. It might be better flying into Jackson WY, or Billings, MT. Yellowstone and Grand Teton NP's are worth at least 5-6 days on their own. I don't know how long it would take to drive to Mount Rushmore.
When we moved to CO from the Boston area, it did take us around 4 days of driving at approx 4-500 miles a day. There really was't all that much worth seeing on the route we took, unless you took detours. I think it was I 80 we took for the most part.