Do NOT try to cram it all into a week!

I hope you have lots of time, because all these spots are worth at least a couple of days. At Least.

I would recommend staying in Yellowstone. Yes, it will cost more, but you're right there. It is a beautiful park.

Take lots of pictures and have a ball...

Where you stay depends on what you can afford to pay. Highest prices are within the park and the lowest are on east side several miles outside the park. Do some research on motel booking sites to find the best fit for you. We stayed in the town of Yellowstone just outside the west entrance to the park and the town is a quirky place and wish there was more time to explore it. We spent only one day there and it was not enough. If I had known that the entrance fee to the park was good for several days or maybe a week, don't quite remember which, things would have been planned differently. Also, the entrance fee was also good for at least another park, not sure but I think it was the Grand Tetons. Check the park website for more and better details.

What you have listed as places to go would be too much to do in a week and seems like at least 2 weeks or more of exploring would be better and more relaxing unless you want to rush from place to place and through each place you visit.

PS: Yellowstone was amazing, beautiful, and didn't have a life extinction event while we were there.

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!

Mt. Rushmore doesn't take a lot of time--an hour or so.

Badlands--okay to drive through. Scenery is pretty. We have camped there a few times.

The Black Hills are very nice to see too--buffalo can be spotted.You will go through them to get to Mt. Rushmore.

Devil's Tower--do not miss! It is great. You can walk all around it on paths, and watch the climbers if you don't do any climbing yourself.

Yellowstone: you can easily spend a week there. We extended our stay there and canceled another park just to stay longer.

We drove south through the Grand Tetons, then headed home (east).

We mostly camped on our three trips out there. National Parks have a lot to offer for campers, but not electrical hookups, at least the last time we were there.

Once when I was a kid about 11 years old we went in July. Never forgot that amazing trip and the things we saw there!!

Then in Sept. 2005, my husband and I went there with my niece and her husband.

Many good tips already given by others but here are a few things I can add. Yellowstone Park is very large. Don't be deceived by the maps. Just driving from one area to another can take a quite a bit of time. So quick trips are not practical. Enjoy the scenery and don't get in a hurry. Also have contingency plans for when your itinerary gets side tracked. What happened to us was that we woke up one morning and it was snowing like mad!! The road going to the northern part of the park was closed and so we had to regroup our plans for that day. It was not a big deal since there are so many things to see and do. Just know that weather or other factors might influence your daily activities.

One thing we did that was fun and not too expensive was get a lunch at one of the General Stores in the Park. We stopped by the deli there and got sandwiches and snacks that we took on an impromptu picnic.

After the trip was over, my husband told me that he preferred the Grand Tetons, which was the second part of our trip that year. It looks entirely different than Yellowstone, although they are very close to each other. The mountains there are very rugged and you will see glaciers and snow atop many of the peaks. While we were there we went on a Photo Op tour that started at the crack of dawn. It was well worth the wait to see the sun shining on the mountains in the distance. There were many other photographers out there, like us, waiting in the chilly air for the perfect moment to snap pictures!

Since you'll be going in the summer, be sure you have your reservations made well in advance so you don't get there and find out that there literally is NO ROOM in the INN!! Oh and for sheer cool factor, if you can swing it, do try to stay at the Old Faithful Inn. There are a variety of rooms available there with a big price range. It is a neat old building and the Old Faithful Geyser is right near by.

Have fun and enjoy planning and going on this trip!

that is a lot to cram into a week and a lot lot of driving. it'll be a day on the road to get from wherever you are in wyoming to the four corners.

if your route to the four corners takes you past mesa verde, it is definitely worth a stop.

edit: i saw you're taking two weeks, should be a good amount of time. the black canyon of the gunnison in montrose, co might be out of your way, but it is a beautiful sight.