I agree with Dave on the routing, but will add a few things you might consider. If you go up US-89 from Flagstaff to the east entrance to Grand Canyon, You might want to include the loop road through Sunset Crater and Wupatki Monument. It only takes you about 30 miles out of the way.

Also, don't miss the Mariposa Grove Redwoods just west of Yosemite. Use the south exit to the town of Mariposa to see it. If you do miss it, you can still go to Muir Woods above SF in Marin County, another redwood grove.

Take the PCH all the way to San Diego and then drive over I-8 to Phoenix instead of I-10 from Los Angeles. It is a much more interesting and scenic road.