High-clearance 4WD vehicles allow you the most access, since many roads are unpaved and only occasionally bladed. (Most off-the-lot SUVs and light trucks are not high-clearance vehicles.) Heed all warnings about road conditions. Remember to buy gasoline whenever you see it.
Some outfitters in towns around the park have hiker shuttles or 4WD rentals. However, if you plan to do a lot of back road exploring, rent a car. Those arriving via Las Vegas will find it the cheapest rental base in the region.