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Getting to Rapid City, I'd take the I-90 route and stop in Rochester. I personally like the drive better, but more importantly its much closer to the halfway mark. Stopping in Des Moines would mean a real difficult drive of nearly 650 miles on day two. However, either way, you're going to be on the road for 10+ hours both days, and while it's not impossible, its a harder drive than we recommend when traveling with children and pets. You won't have time to do anything other than drive and make the minimum stops for gas and restrooms.

Having said that, your following days really need to be worked on. Day 4 and 5 are close to impossible. When you factor the extremely slow drives through Yellowstone and the Tetons. They are more days where all you'll be doing is driving, driving, and driving, barely having time to even get out and take a picture. Keep in mind, Yellowstone is a huge place, where just to take a quick look at the highlights of park takes a pretty full 2 days.

I think if you want your family to have a good time, Grand Teton really has to be thrown out of the picture, and really, I'd try to cut down your drives in general so you have time to actually get out of the car and have some fun!