I chose to start at the Western end of my trip. Cedar City is good sized town with plenty of restaurants and places to stay. Getting there is your choice. Salt Lake City and Las Vegas are both within a reasonable driving distance. I choose Denver for flight availabilty and because I love driving through Colorado.
First day was Zion Canyon National Park. No cars allowed in most of it so you have to take the park shuttle up the access road. It's a great way to see the park with several stops at sites and trailheads. You should hike into the Narrows as far as you can where the massive walls close in as you go up the river. On the slightly challenging Emerald Pools Trail you can hike behind some narrow falls. Many more challenging hikes are available including some to the top of the canyon.
Leave the Park through the east on the Mt Carmel Highway - 9 and climb to a remarkable 1.1 mile tunnel with a view that you won't forget. Continue on this highway for several miles to see wildlife and more fantastic rock formations including the Checkerboard Mesa. You have choice to loop back to Cedar City and on the way and visit Cedar Breaks National Monument.
The next day I headed east to Bryce Canyon N. P. You don't need to drive the 19 miles into the park to see its best feature. The Amphitheater (near the entrance) is the jewel of this park. A giant opening with thousands of rock spires called "hoodoos", faults and rock uplifts is truly unique. The Navaho Loop Trail is listed as "moderately difficult" by the NPS. But it is not for weak and lazy. The 1.7 mile loop goes straight down and straight up on numerous switchbacks into a slot canyon. Somewhat challenging for flatlanders because it is at 8300 ft. above sea level. Well worth the effort and the best way get "into" the Canyon in a day visit. The Rim Trail is level and has many great photo views also.
Now if you prefer to drive like I do, the next hundred miles is indescribable. Highway 12 east has cliffs, faults, monoliths (solid rock mountains), bowls, canyons, and rifts on a twisty two lane. You will end by climbing to just under 10,000 feet into the Dixie National Forest and can stay in a small town called Torrey.
Next morning you are minutes away from Capitol Reef National Park. This Park is a hikers dream and you can go as far as your legs will carry you on many backcountry trails. If you camp, the Fruita Oasis is beautiful.
A short hike to Hickman Natural Bridge provides great views of rock formations like Capital and Navajo Domes. Hiking the Capital Gorge area is level but also gives a great sense for the immensity of the Reef area. Signs warn about flash floods. So ANY hint of rain means keep out of the canyons of Utah.
The long drive to Moab, which a base for Canyonlands N.P and Arches N.P., has opportunities for many side trips and stops. Lake Powell, Glenn Canyon, Monument Valley and Hovenweep National Monument are within reach if you enjoy the drive around this picturesque part of the country. A day in Natural Bridges National Monument is easily accessible and worth every minute. If you have little time at least drive the park loop for great vistas. If you have more time a moderate hike to Sipapu Bridge is worth all the narrow cliffs, steps, ladders and rock scrambling handrails to the view on the open rock ledges. There are two somewhat easier bridges to hike to but the park is at an altitude of about 6500 ft.
Next, move up through Blandings and perhaps elect to see the southern, or Needles area of Canyonlands N.P. Moab is a tourist hub and a great place to stay for the Northern end of Canyonlands (Island in the Sky). A stop at to Mesa Arch shortly after sunrise will get you a classic photo-op and the views of the open canyon at Grand View and Buck's Canyon Overlooks display all the different landscape levels. Hike the Upheaval Dome on a short overlook trail or longer around the dome.
At least a full day is required to take in Arches National Park, which is adjacent to Moab. Everyone raves about the rocks in Sedona, but the formations, arches, spires and balanced rocks in Arches are like no place you've seen. Plenty of hikes, both mild and challenging, can provide a great activity to see the sites of the park. Recommend Landscape, Delicate, and Rotunda Arches, Balanced Rock and the Windows. In the same area is the fantastic Double Arch which was a location for a young Indiana Jones in the Last Crusade film.