Best time to see this great part of the country is May or Sept./October. Arches, Canyonlands, Zion can get very hot in June-Sept. My suggestions coming from the Eastern U.S. would be:
I-70 to Grand Junction, CO (side trip to Colorado National Monument)
Arches National Park (2 days)
Canyonlands NP (1 day)
Monument Valley (1 day)
Canyon DeChelley National Monument (2 days-day trip in canyon )
Antelope Canyon near Page, AZ
North Rim of Grand Canyon (2 days)
Zion NP (2 days)
Bryce NP (2 days)
Drive Utah 12 (Hike to Lower Calf Creek Falls)
Capital Reef NP (2 days) If you can dry camp stay in Fruita CG.
Return home.
Let me know if I can help.