From Flagstaff to Salt Lake City will give you some nice drives if you stay off interstates. However, you won't see 'all' that Utah offers.
You want to take 89 north out of Flag to Page, AZ and continue to Kanab, Utah (Best Friends Animal Sanctuary is here - an awesome place - sign up for a tour) There's also Coral Pink Sands State Park with colored sand like you've never seen before. You could also drive to the North Rim of the Grand Canyon from Kanab as a day trip.
89 north to UT9 will take you into Zion Nat'l Park. If you want to stay in Kanab area and just do Zion as a day trip, you can make it a very scenic circular drive. Hwy 9 east into Zion and the towns of Springdale and Hurricane then pick up UT59 which becomes AZ389 taking you to Fredonia at Hwy 89. Then 89 north to Kanab.
89 north to Hwy 12 east will take you to Bryce Nat'l Park and on to Torrey, UT and Capitol Reef Nat'l Park.
From Capitol Reef that 24 to I-70 a short distance to Hwy 191 south to Arches/Canyonlands Nat'l Parks.
Then backtrack on I-70 to Green River and take 191/6 north bringing you to I-15 at Spanish Fork, UT just south of Salt Lake City.
All roads are RV-friendly. We've driven them many times with our 40' motorhome towing the Jeep. They are popular RVing routes.
Have a great trip!
Full-timed for 16 Years . . .
Traveled 8 yr in a 2004 Newmar Dutch Star 40' Diesel
& 8 yr in a 33' Travel Supreme 5th wheel