Gemini Bridges is a great drive and incredibly basic. The back way into Dead Horse Point through Schafer Canyon is also a dirt road with great views and a fun set of switchbacks with a drop that can be unnerving if you're sensitive to heights, but is a real fun family drive.
If you're looking for actual wheeling but don't know where to start, I'd run Chicken Corners and Fins and Things. Chicken Corners is
almost technical wheeling, I would call it a 3 out of 10 (you wouldn't want to take a low-clearance car out there, and you might use 4wd unless you're good at picking lines). Fins and Things is the iconic Moab experience, and can be driven by a novice driver in a stock truck if they're careful - it's got lots of VERY steep climbs and descents that will have your balls crawling up your throat if you're a noob, but the slickrock gives awesome traction and there's not too much that will tip you sideways, so it's a good trail to check yourself on. If you haven't driven on slickrock before, you'll be amazed at what your truck can go up. I'd call it a solid 4/10 (high clearance required, you WILL need 4wd, and you'll want low range in a couple spots).
Traildamage.com has ratings and directions/GPS for both those trails, or hit up a local tourist trap for a copy of Charles Wells'
Guide To Moab UT Backroads And 4WD Trails,
looks like this and you'll find it in most stores along the main drag through town. Either one of those are great resources and will help you get the most out of your time down there. Run those two trails and use your experience on them as a measuring stick against the ratings in that book and on Traildamage, and you'll have a good idea of where else will be fun.
Carry lots of water, it's hot down there in case you hadn't noticed.
