A Workable Route and Other Suggestions
One basic, but very scenic, route that would link Arches, the Grand Canyon and Yosemite is to take I-70 out of Kansas City through Denver and on to Utah where US-191 would take you south past Arches and down to US-163/US-160/US-89 to the Grand Canyon. After driving along the Rim Road (AZ-64) continue down to Williams AZ and I-40. You then have a choice of taking US-93 up to Las Vegas then through Death Valley to US-395 and over Tioga Pass (CA-120) into Yosemite, or continuing on I-40 to Barstow where you'd use CA-58 to Bakersfield and CA-99/Ca-41 up to Yosemite. Finally, CA-120/I-80 would take you into San Francisco to finish up your move.
Now if you take the above route, you'll also be passing near enough to several other great places that you could visit as time permits, including Rocky Mountain National Park, Colorado National Monument, Canyonlands National Park, Goosenecks State Park (Utah), Monument Valley Navajo Tribal Park, Wupatki National Monument, and either Mojave National Preserve or Death Valley.
AZBuck