We did a driving trip like this when our kids were young teens -- it was a fantastic trip!
Lots of good information on RoadTripAmerica.com, and lots of good information on the national park websites.
Look into an America the Beautiful pass. For one price, you can enter all the national parks for one year. It may or may not be worthwhile for you, but it saved us money, especially because I was able to lend it to a friend for her trip to the Grand Canyon (you're allowed to have two names on the pass) and because I used it the next summer (literally the week before it would've expired) in Puerto Rico. Don't buy it until you reach your first park; your year begins when you buy it. And add up your costs ahead of time, recognizing that not all parks use the same fee schedule; for example, Mt. Rushmore charges per car . . . while admission to Yellowstone ALSO gets you into Grand Tetons . . . and Wind Cave is free, but you pay if you go on a tour of the cave . . . and Great Smokies is altogether free -- you must do the math for yourself to see if the pass is worthwhile.
Definitely take a big cooler and plenty of water and juice drinks; in the dry heat you'll want drinks constantly, and sodas just don't go down particularly well. We always had sandwich fixings, fruit and yogurt in the cooler, and we ate many roadside meals.
My personal favorite stops:
- Mesa Verde National Park -- We took two ranger-led hikes to see the old Indian villages built into the hillsides; absolutely amazing, and you can't go without a ranger (each tour costs only a few dollars). Note that one of my children had some trouble with altitude sickness. The negative: This is close to nothing else.
- Antelope Canyon -- Absolutely amazing. Otherworldly. Because it's on Navaho land, you must go with their guides. Even if that weren't a requirement, you'd want to do it -- a couple of the big trucks got stuck in the sand on the long drive out to the canyon. I'm 100% certain our little rented Saturn Vue (?) wouldn't have made it. In this same area, we visited a dam (second biggest, but I forget its name); my husband was thrilled.
- Zion National Park -- Wow, the hiking! So beautiful!
- Grand Canyon -- Wonderful, wonderful, but not the absolute best we saw. Oddly enough, not what I expected. We were at the North rim over July 4, and the parade they threw (which morphed into a large-scale water fight) was one of my kids favorite parts of the whole trip.
- Yellowstone -- So diverse. So beautiful. I absolutely want to go back. We did the cookout experience, which was really fun, though pricey. Look into the ranger programs. Take the hike up to the mountaintop and see Old Faithful from that vantage point. Very crowded in the summer, and people cannot drive. Do not enter the park without a FULL gas tank; they sell gas inside the parks, but oooh the prices. Do not take risks with food -- bears will win every time; they can even tear a door off your car to get a loaf of bread. Also take jackets; we were there in July, and I ended up shelling out $50/person for sweatshirts because the temp dipped unexpectedly to 50 degrees!
-- Custer State Park -- It's a state park, not a national park, but -- WOW -- was it great. Possibly our #1 stop. Their claim to fame is buffalo. We did the buffalo safari, which I highly recommend -- though, again, it was expensive. We were the only family on the tour, and our guide was an elderly man who really knew his stuff. He took us into parts of the park not accessible to tourists and told us great stories. He also left us with a healthy respect for how dangerous buffalo are. I want to return in the month of September, when they herd the buffalo every year.
- The Jenny Lake hike in Grand Tetons. It was the single most beautiful hike I've ever taken. We took the ferry across the lake instead of hiking alllll the way around. The view from the top was incredible. In fact, everything in Grand Tetons was incredible. We did an evening rafting trip down the Snake River, which was enjoyable -- but it was something you could excuse yourself for missing.
We were gone for 3 weeks and saw SO MUCH . . . but these are the big stand-outs.