Good old Sandy Balls in the new forest. Loads of cycle routes from here, the bike hire shop on site will even sell you some routes (or they are free if you hire their bikes).
Loads of routes to choose from, or you can take your bikes out on your car and there are loads more routes within a half hours drive. A good one is Moors Valley, a massive country park in the Verwood forest, several routes around here as well as a nature trail, some lakes, and for the kids a minature steam railway.