How many days are you planning? Do the slickrock, hell's revenge, maybe porcupine rim / la sal mtn range and Arches Nat Park is sweet. There are several legal turn offs in the park where you can wheel back off the paved road to see cool stuff.
Been there three times but never in a vehicle. Once on a mtn bike, and twice on dirt bikes.
On the west side of town are some other famous trails but we didn't have time to do both east and west.
Great place to wheel... campin' at any of the BLM campgrounds is cheap, you shouldn't have any problems with crowds that time of year. I was there this past April right after jeep week and it was a zoo. Also there June and it was freakin' hot, 115 on trail. If your campin and want to do a backcountry trip run BeefBasin Rd into the south entrance for the Needles then out on Elephant Hill. We did that this summer and it was a blast. If interested I have GPS for that trip. Also Lockhart Basin is a great overnight trip. For eats ZAK's wood fired pizza is good as well as Pasta J's... If you want a local to wheel with I have a buddy that we stay with who lives there. It's amazing what the locals can crawl up in their rigs. I rode with him one day and he scared the shit out of me. Anyway, if you want more specific info PM me...