Assuming you will be coming along I-70 from Colorado into Utah, then I-15 south into southern Utah and Nevada. Moab is about 41 miles south of I-70 as you enter Utah from Colorado. Lots of trails and other routes in the Moab area. I would recommend some mild, not hard core routes in and around Moab such as: Onion Creek, the Shafter Road, Potash Road, Harrah Pass.
When back on I-70 heading west thru central Utah, you could visit a number of mild roads north or south of I-70 in the San Rafael Swell area. Its beautiful country and really ought to be a national park. The Wedge Overlook is awesome, viewing the "Little Grand Canyon". Buckhorn Wash is awesome, with lots of Native American pictographs (please help preserve them). South of I-70 along here is Goblin Valley State Park with lots of crazy hoodoo rock formations.
If heading back home via Arizona, north of I-10, heading up I-17 for a ways is Sedona, AZ. There are scenic rock formations there, as well as a very mild (the most mild?) Jeep Badge of Honor trail named Schnebly Hill. You would give your family a taste of off-roading there for sure.
Here's a map link: http://www.castlecountry.com/media/u...-Swell-Map.pdf
Once on I-15 heading south toward Nevada is a main entrance to Zion National Park. Not off-roading, all paved, but well worth a visit while in the area.
Once into Nevada, just west of Las Vegas is Red Rock Canyon which is a good stop. Crossing into California from Nevada is the Mojave National Conservation Area with lots of interesting desert flora such as the Joshua tree....
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