There are almost 100km of bicycle trails along 5 river banks in the city, including out to the coast just past Danshui/Tamshui.
Unfortunately I didn’t get to do any bicycling in my limited time; the one day I tried, I found out I couldn’t rent a bicycle at one location and return it at another (as I had been misinformed at the tourist office).
For a detailed bicyling map see www.taipeitravel.net/download.do?fileName=/d_upload_ttn/att/files/10002736.pdf. (5 MB).
There is also a free short-term bicycle hire system called U-bike. Currently it’s only in Xinyi district (i.e. between Taipei City Hall/Sun Ya-sen Memorial Hall and Taipei 101). However, you have to have a smart chip credit card.
See www.taipeitravel.net/en/ for general Taipei visitor info.