You will have a great trip but some places depend on the ages of children. Your first three are well worth seeing but simply can't compare with Yellowstone and Teton National Parks.
They get VERY crowded in June; try to do good research on accommodations that you can afford and can get reservations for. In Yellowstone the area up at Mammoth Hot Springs is usually less crowded and camping is great. Also in the Tetons. Bypass Jackson Hole, crowded and touristy. Less expensive out of the Parks.
The four corners has so much to offer that you really need lots of time to see it all; you might want to just plan on one site like Chaco Canyon or Aztec. Magical if children are old enough to appreciate the ruins. Difficult to get in and out of Chaco; no overnite accommodations except camping. I would recommend the Ute Mountain Tribal Park, even over Mesa Verde, because it is as if the ancient ones have just left. It takes a full day and is a guided tour; you must have reservations. Santa Fe is great too but crowded and expensive in summer. It will be more interesting if you have time to research what you'll be seeing. And if you love mountains as I do you won't find any place more beautiful on earth than Telluride, Cortez, and the Sangre de Cristos in Colorado. Come back often!
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