I used to live in Kanab, Utah (bottom left corner), and Utah is a favorite place of mine. .
Utah is a huge state with a vast amount of public land. There are trails everywhere. Rock crawling competition trails in Cedar City; backroads like hole-in-the-rock thru the Grand Staircase, gravel roads that lead to hidden canyons. Sand dunes near Kanab. All sorts of stuff everywhere in the state.
There are also some fantastic national and state parks that might not require 4x4, but are worth the visit.
Traildamage.com has a nice rundown of a bunch, but more Moab centered. Charles Wells has a book on the state.
If you're looking for adventure, you could do a very lonely, quiet trip into the Grand Staircase Escalante or into the maze section of Canyonlands - 4x4 very useful but not super tough trails. Then, on to the more populated and very tough trails around Moab.
Read Edward Abbey first -- Desert Solitaire and the Monkey Wrench Gang. It'll make te desert really come to life.
There are lots of adventures in that state.