Although we live in such a great place for walking with kids, it's not always easy to find safe cycle paths where they can really take off without the worry of traffic. There is Dalby Forest which has a good family cycle trail plus a lovely cafe to have refreshments in afterwards.
A couple of weekends ago we tried out the Solar System Cycle Ride in York which runs along the old East Coast mainline railway. The ride is a 12-mile round trip and starts at the sun, ending at Pluto. The distances between the planets is to scale so the gap between some is smaller than others. The boys really enjoyed it. There is a bit of information at each planet too.
We parked our car at Askham Bar park and ride and there is a cycle lane just across the road that leads you to the cycle trail. There are pubs on route for refreshments and there is also a little cafe at Naburn although you would need to check for opening times in the autumn and winter months.