Taipei City bicycle paths map
I came across this map of Taipei’s bike paths in this thread at Forumosa. It shows that there is a quite extensive network of bike paths around Taipei. They mostly (but not exclusively) follow the rivers that go through Taipei. While they might not be very practical for commuting they would be good for weekend riding.
Many MRT stations provide convenient access to some of the paths. These MRT stations are clearly marked on the map. Conditions apply to taking bikes on the MRT. In summary they are only at certain stations on the and only on weekends and public holidays before 4:00pm and after 7:00pm. You can pick up a brochure in English from any MRT station for more details.
Update: Click on the map above to see a large version of the map. You can also check the Taipei County bike path map (newer version) or download the MRT’s Guide to Hiking and Cycling (5MB pdf file). If you have a link to any other maps please leave a comment. The Taipei MRT have also changed the regulations to make it easier to take bikes on the MRT. The Taipei MRT website has the detailed regulations and I have updated this post with a brief summary.