Matts Offroad is in St. George, not Moab, but there are recovery guys in Moab too just not as internet famous.
A longer day trip is going up Gemini Bridges, then hit the highway and go out to Dead Horse Point, then on the return take Long Canyon. If you are into Mountain Biking there are some really fun trails in Dead Horse Point now, very family friendly. They do rent bikes out there too.
We like going out to Tusher Tunnel. The SxS's have really torn the trail up out to here but it is still very doable in the expedition. I did it this spring. My FIL had his brand new Ram 1500 Rebel and he was freaking out a little. It was over his comfort level. In years past I have taken people out there that were in Honda Pilots but I don't think I would do that now with how bad the trail has gotten.
Not an offroad trail but the Corona Arch hike is always a family favorite.
If you want to drive a little bit, roughly an hourish from Moab is Goblin Valley and this is also a family favorite. My kids can play here for hours. If you are into hiking look for the hike to the Goblin's Lair, not super long hike either.
Its been a long time (we mountain bike more than 4x4 now) but I think Onion Creek and Fisher Towers is easy.
Years and years ago I bought a book called "Guide to Moab, Ut Backroads and 4 Wheeldrive Trails. It is always in the trailer for when we go to Moab. We go to Moab at least once a year, sometimes twice a year.