Go south to Tharwa and try the road to Fitz's Hill... you'll either want to turn back after it, or keep going. Pleasant country. If you keep going, you'll get to Namadgi National Park which also is worth exploring, but the roads start to turn to gravel.
Canberra is serviced by an extensive network of bike paths that will allow you to explore to the satellite suburbs. Plus you can ride around Lake Burley-Griffen which was pleasant when I did because the environment changes a bit from urban to forest at one end.
I'm not sure, but I think the Brindabella area to the north-east also could be interesting, although the fires several years ago may have taken away some of the scenic amenity, and I am not sure if the roads are sealed.