Do a search here on DRN, Michigan Trail Riders section for Pinecone, Spring Pine Cone, or Mini Moose - you will get a real good feel of what there is! There is an entire series of races exactly like what you describe, called the Family Enduro Series. The next event in FES is the Mini Moose in MArquette on 7/29. After that, I think there is one in Indiana. You just barely missed the Pinecone last weekend. Earlier this year was the Spring Pine Cone, and the Sand Booger.
The trails in the FES series are not ultra gnarly difficult stuff like the full tilt events, but usually do include some "event trail" which is way tighter than any public trail stuff - but still entirely rideable by a younger or less experienced rider. These events are the perfect way to learn about doing enduros, the ins and outs of timekeepin etc! Sit tight here - I am sure Fatherandson, and a few others directly involved in putting on the FES will add much more info!