If you just want some easy off roading, head west towards Black Mountain. You can camp, shoot, drive around on rough roads to get your feet wet, etc. I take the family there sometimes for early season, quick "have to get out" weekends when I don't want to risk breaking anything. There are a lot of canyon roads that take you partly up a canyon, and we will often hike to the tops, or to geocaches, or just get out for some rock scrambling. Some of the canyon roads will take you through the mountain range, but most will dead end. From the exit with the gas station (the only one out here) head north, and find a road that crosses the RR tracks. From there, head a bit NW so that you will stay on the west side of the mountain range. There are a few rough roads that skirt the south end of this area, and there is a handful of geocaches (if you like that sort of thing) but otherwise just your average jeep road. If you stick to the roads and canyons on the west side of this range, you will avoid most of the pollution from the magnesium factory. If you are on the east side, there is a landfill and the mag factory, which smells strongly of chlorine. Yuck. Too bad too, some nice canyons on that side.
http://g.co/maps/m7jbt
If you skim google maps using satellite imagery, you can see a lot of roads headed south from the same area off the freeway, and they look like good candidates too, but I haven't gotten down there yet, so I can't vouch for it or suggest anything.
If you want to go a bit further south, over at 5 mile off road area, there is a lot of roads that are challenging but not terrifying for the new wheeler. One other popular area starts on Sunshine Canyon Road, south of Tooele. Most get there by way of Eagle Mountain though. You can find just about anything you are looking for there. Mild rock crawling or plain rocky dusty trails. Good views from the tops of ridges and small mountains. We usually camp near the bottom, but I think you can camp up farther too. Not so far away you have to camp, but nice place to get a weekend worth of wheeling in.
http://g.co/maps/qg2fu
Again, scour google maps using the satellite imagery, and you will find a lot of possibilities in all directions.
American Fork Canyon is great, and on my second visit, I bought an annual pass, and I will continue doing that because it's an incredible value, I have used it all over Utah, and gotten my money's worth very quickly. There is a lot of places you can go in a stock Liberty up there past Tibble Fork Reservoir, but use caution because some will look average and then turn ugly fast. You will wind up with dented rockers, punctured pans, damaged gas tank, etc very easily. A road will turn from rough to ROUGH pretty quickly. A lot of folks go up in stock small 4x4s and get a lesson in prudence.
If you have questions about a specific area, post them up because there is a wealth of knowledge here, on tap.