Bikes Permitted on MRT Weekend and Holiday Evenings
I haven't seen this mentioned anywhere in the English media, but you can now take your bicycle on the MRT on weekend and holiday evenings. This makes later afternoon riding in Xindian or Danshui much more practical. Remember, you can only take your bike on the Danshui-Xiandian or Beitou-Nanshijiao lines. Here's the official announcement.
Also, on returning from a ride on the Northern Cross Highway recently, I was able to take my disassembled bike in a bike bag on the MRT from the Yongning station in Tucheng back to Taipei on a Monday morning around 10am. The MRT regulations say that they can refuse passengers with oversize luggage, but they simply asked me to use the elevator. This is another option that makes all the great riding in the Sanxia area more accessible from Taipei.
Of course you can always ride out to Tucheng on the the bike path along the Dahan River.