Hi Jack,
I assume you'll be coming down through Salt Lake on I15. If so, take highway 6 out of Provo, through Price then down to Moab. If you're looking to stretch your legs in Price, they have a great dinosaur museum at the College of Eastern Utah. In Arches, do the out-and-back drive to the end of the road at Devil's Garden. Along the way, stop at the Windows section for some short hikes to nice arches. I highly recommend the hike up to Delicate Arch; it's the iconic arch of Utah. This picture shows the big slickrock bowl below it:
In Devil's Garden, it's not too far to see Landscape Arch, the longest arch in the park. If you're up for a bit more adventure, sign up for the Fiery Furnace hike. It's a ranger-guided hike, not really all that long distance-wise, but may be a bit more strenuous because you're scrambling between slickrock fins.
The Island in the Sky district of Canyonlands has dramatic overlooks. Drive out to the Grand View Point Overlook. Along the way, stop to check out the Shafer Canyon overlook and see the switchbacking road going down into the canyon (and think about crazy people like me riding my mountain bike UP it). Mesa Arch is a short hike that ends at a very scenic arch along the canyon cliff edge.
As you drive through Capitol Reef, stop to see the petroglyphs along the road. The trailhead to Hickman Bridge is also right along the road and a nice hike.
Highway 12 is incredibly scenic and has a lot of variety, climbing high onto Boulder Mountain, then dropping into deep red rock canyons. There are some pretty steep grades in a few places, so gear down, take your time, and enjoy the view.
Sunrise, Sunset and Bryce Points at Bryce Canyon are all great (these are scenic overlooks). If you want to try a hike down amongst the hoodoos (rock spires), the Wall Street/Navajo loop trail is way cool.
If you're heading to Zion from Bryce, go into Zion on the east entrance. I love the slickrock domes on the east side. The Canyon Overlook trail just before the mile-long tunnel is a nice hike that leads to a great view of Zion Canyon. In Zion Canyon itself, the Emerald Pools hike takes you to waterfalls cascading off the cliffs into pools. The Riverside walk trail at the end of the Zion Canyon road takes you up to the start of the Narrows, where sheer red-rock cliffs rise about 1000' up.
There are so many great things to see and do in southern Utah! Have fun!
Jeanette