Bedgebury is much improved recently, but a bit of a hike if you’re only down for the weekend. Occasionally Bedgebury gets so busy they have to turn people away, so beware.
It is now possible to cycle the length of the river Medway from Aylesford to Teston (having said that you can’t at the moment from Maidstone East up to the new bridge in town – this can be navigated on road, but not ideal). You could start from town, however, and ride upstream to Teston. At the Teston end is a country park, which is quite pleasant.
You could have a ride around Mote Park, takes around 45 minutes with smallish kids. I think they have been working on the pump track there, so that might be fun, plus there’s a play area and somewhere for ice creams, if your kids are of that age.
Let me know if you need any other info