See the links below for details of rides that can be done as day trips from Taipei and further afield. They will take you into the mountains where the clean air and beautiful scenery will make you feel like Taipei is a million miles away.
Bikes can be taken on the MRT in Taipei although conditions apply. In summary only on weekends and public holidays and only at certain stations. Pick up a leaflet in English from any MRT station for more details or check the official regulations on the MRT website. This can be a convenient way to access the network of bike paths along the rivers in Taipei. A map of the paths in Taipei County can be found here. You can also download the Taipei Metro’s Guide to Hiking and Cycling (5MB pdf file).
It is also possible to take bikes on trains in Taiwan. Taiwan in Cycles has compiled detailed information in English about the Taiwan Railway Administration bicycle policy.
It is possible to hire bikes at many locations around Taiwan. There are bike hire stations at most places where bike paths have been built. For example, in Taipei they can be found at numerous places along the riverside bike paths including Bali, Guandu, Dadaocheng, Gongguan and Xindian. If you want to hire a high quality bike for touring then you can rent a Giant from one of twenty stores around Taiwan. More details here. Alan's Mountain Bike in Taipei also rent quality mountain bikes and organise rides on weekends.
This Taipei Times article contains information about many of the bicycle paths around Taiwan. Have a look at Rides and Trails on the Formosan Fat Tire Association's web site. It has lots of information about some off-road trails around Taipei as well as rides elsewhere in Taiwan. There are dozens of roads and trails offering great riding so just get out there and explore!