Having done a round trip from Oxford, Oh, (west side) to Yellowstone every summer for over 18 years, It's hard to tell you how long it will take. We could make it in 3 long days of driving, but as our vehicles got better over the years we would add a day or two to sightsee. If you're stopping at the Badlands, you could stay in Wall, SD, next to I-90. I'd also swing by Mt. Rushmore, (walk the trail across the front of the mountain to the Artist Studio) and even more interesting is Chief Crazy Horse. (They have a very nice Indian Museum there) In Wyoming, I'd enter YNP through the East Entrance from Cody, Wy. The Buffalo Bill Center of the West, in Cody Wyoming
is worth a visit. In Yellowstone, be sure to get out and hike a bit. The hike up Mt. Washburn is a wonderful walk in the park. The Visitor's Centers at Old Faithful, Canyon, and Mammoth have great exhibits explaining much about YNP. If you go south to Grand Teton NP, the Visitor's Center at Moose, and Colter's Bay, are also very nice. Some good hikes in GTNP are Hidden Falls/Inspiration Point among others.
So... You could spend a couple of weeks doing some of those things. To in addition add the Grand Canyon and perhaps some of the other great NP in Utah, (Arches, Zion, Dinosaur National Monument) you're really overflowing the plate!
Many years ago I took a group of 20 or so young teens on a backpacking trip to Banff and the Canadian Rockies. We hiked for 10 days, then drove down to YNP, Mt. Rushmore, and more. By the time we got to the Badlands, the kids were so burned out sightseeing they just watched from the bus as we got out and looked around, read signs and took photos. I learned to scale back how much to visit on any one trip. If they enjoy the first trip, you can always take them out again.