Maybe it would help to know what you are interested in. We like to hike and explore ruins in the less crowded (and often more remote) areas. In addition we like to have our dog with us which tends to put us on BLM, National Forest, and National Monument land. So our usual haunts are the Escalante area, Arizona Strip, San Raffel Swell, Comb Ridge/Cedar Mesa, and Moab areas. There are great places in any of these locations to set up a base camp from which to explore. But first you have to decide what you are interested in (i.e.; the ruins tend to concentrate in the SE corner of the state where the best slickrock hiking tends to be a little further north. Exploring these areas tends to take a bit of adventure as there are few marked (or even identified) trails. But for those who have the skill and willingness to explore, the rewards are great.
If, on the other hand, you are into National Parks (and all their accompanying rules), there are plenty of those in Utah as well. Trails in the parks will be better marked ..... and traveled.
I suggest you start with this
book.
Hope this helps,
Steve