I cycle whenever I can. In Jan 2010, Luke and I cycled through Bangkok and Ayutthaya in Thailand. In 2009, I cycled in Amsterdam. I really enjoyed cycling and whenever the opportunity arise, I like to travel by bike.
We got off at the Danshui Station which is the last stop on the Red Metro Line. We don't know where to go to rent bikes then so we asked the Info Desk at the station. Good thing that he can speak English and there is a bike rental 5 minutes away from the station. To be honest, I just do my research but I don't do enough. Good thing I am lucky with life. LOL!
We rented our bike for 2 hours for 150NTD each. BTW, if you want to rent bikes, you should bring a proper ID with you as it is a must. For 2 hours, we cycled for 12 KM. I think that's a good achievement. It was fun cycling from Danshui to Taipei but from Taipei to Danshui is a bit of hell. We were both hot(we started to cycle at 1PM) and thirsty. The view gets uglier and uglier the nearer you get to Danshui Station too. I am just being honest here.
If you are interested to do your own cycling tour, this website may be of help - http://www.taipeitravel.net/frontsite/en/sceneryEnListAction.do?method=doFindAll&menuId=1030501&typeId=8
There are lots of options to choose from.
Photos from our 2 hour cycling tour: