In my opinion that is simply too much to see in the time you have You'll be rushing to see everything without taking enough time to REALLY see it.
Three places, in particular need to be seen over at least a couple of days, if not more. The Grand Canyon, Bryce Canyon and Yellowstone.
Last year I had a cousin fly in to Phoenix where I picked him up and he drove with me to the Grand Canyon and Bryce. From Phoenix, you'll be going thru Flagstaff. On the way you can make a quick stop at Meteor Crater. It's only a few miles off the Interstate and just East of Flagstaff.
There is so much to see everywhere you're going, especially in Utah.
Mike