If you want to be close to town and still have trails right out the zippered door of your tent you can stay at Sand Flats Recreation Area. Its first come-First serve but spots turn over everyday. You will have access to Fins N things, Hells Revenge, Porcupine Rim and Sand Flats Rd. Its a beautiful area which means its popular and often has crowds.
If you want to be further out of town, about 10 miles north theres a few campgrounds up on the hill towards Canyonlands on Hwy 313. There are lots of "Overland" style trails out there. You will be close to Long Canyon, Shafer Switchbacks, Mineral Point, Tusher Tunnel, Secret Spire and even the Holy Grail that is The White Rim.
South of town gets more limited as there is some dispersed camping but not a lot. This is the area of Strike Ravine, Behind the Rocks, and Picture Frame Arch. There are some hidden gems down there but they are few and far between. (38°22'46.6"N 109°25'20.7"W is a personal favorite ***not little kid friendly -- Cliff edges***)
Fruita to Moab on a Mountain Bike is a real thing.... Its called the Kokopelli Trail. While its cool, Bike-Packing is not my thing.
Hanging out in Moab for about a week maybe exploring other areas. Plans not set. Upstate New York to Moab, June 10-June 22! Would love to meet people from this group and soak up all the information on this amazing sport we can.