Family riding is the Strawberry line, a disused and re-purposed railway line. The tunnel at Winscombe/Kings Wood is closed for work at the moment, but there is a diversion.
All the good riding immediately around cheddar is either cheeky, rocky, or both, with the exception of the Gorge walk, which is classified as a Bridleway. It’s a stern climb up, and a good, if short descent that’s best taken at speed early or late to avoid the hordes. Mind the cliff edge.
The Mendip plateau is close, but from Cheddar, all roads (and indeed, trails) head up. legit Off-road is either Warrens Hill Road (a BW), or ‘Yoghurt’, a bridle way around the back of Cheddar Woods Lodge park. By road it’s a stiff climb to Shipham, or you use the Strawberry line to put some miles into Sandford/Winscombe, where you can go up past Sandford Quarry and into a BW, crossing the A38 bycthe disused Churchill quarry at Dolbury Bottom and climb your way up from there.