How much time have you got?
I would recommend taking Kane Creed Blvd and follow it to Hurrah Pass. From town, turn at the McDonalds and bear left a little way up the road where it forks.
If you like that area, you can go down the road on the other side of the river. As you are leaving town to the north, make a left on 279, which is right after a large construction project and before Arches National Park. Lots of petroglyphs here. You can take it all the way to the end of the White Rim Trail, and do some of that.
Arches National Park shouldn't be missed.
Also on the north side of town, HWY 128 which leads to Castle Valley and is a very scenic drive.
From the middle of town, take 300S East (Phillips 66 gas station) to S400 where it makes a T intersection, turn right (south), then make 2nd left on Mill Creek. Stay to the left where it forks and becomes Sand Flats Rd. You might have to pay a couple bucks to enter, depending on time of day, but it's worth it. You will drive through the slickrock area and see the entrances to some pretty well known trails (none are suitable for a stock pickup). The road eventually gains a good bit of elevation and dead ends with the La Sal Mountain Loop Road. I would recommend going left. Nice Alpine mountain road. When it dead ends, make a left which will take you to Castle Valley, then make another left on HWY128, which will take you along the Colorado River back to town.
All of these routes are paved or improved dirt roads which shouldn't be a problem for any 2wd vehicle with a little bit of clearance. The clearance would only really be needed with a couple of water crossings on Kane Creek. Everything else you can do in a civic.