If you google "jeep tour" or something similar for the nearest town to the area you'd like to explore, you will find some small companies that specialize in guided jeep excursions. There are also companies that rent jeeps.
Having read the above with some interest, I might add that the White Rim and Shafer trails near Moab are not super for anyone with a fear of heights! We did the Shafer trail with a friend-of-a-friend who was from the area, and going with someone familiar with the sights who enjoys explaining them is a definite plus. The road itself was in good condition but it is narrow, dirt, and, well, it's a long way down. There are many other interesting routes in the back-of-beyond around Moab that don't require much more than a high-clearance vehicle.
In the Moab area, the Information Center staff Moab Information Center - Get up-to-date vacation planning information from the official government tourism website for Moab, Utah.
and the Bureau of Land Management field office can recommend routes suitable for your interests, type of vehicle, and abilities.