Although Salt Lake City [1] is well connected with the rest of the world and has a good public transportation system, to really explore the state, a car is the most realistic choice (though a case can be made for a bike and strong legs). However, if you’re planning a winter trip with a focus on skiing the Wasatch ski areas, consider skipping the car rental and relying on shuttles or city buses.
Links:
[1] http://www.moon.com/destinations/utah/salt-lake-city