Many Canberra nature reserves have been reviewed on Australian Hiker.
Cycling is permitted in 30 of the Canberra Nature Park reserves. Information from the Management Plan 2019 can be found in OpenStreetMap. The rideable reserves are previewed here.
Crace offers much: a bike track looping the suburb, a magnificent view from Percival Hill, and the picturesque Gungahlin Pond. You can ride to all with only one quite road crossing, so it is ideal for the kids.