To get in touch with several mountaineering clubs at once, you have to go to the Taipei Main Station. Severals clubs meet there, at the West #1 exit, on weekend mornings between 7:00-8:00, and each club has a different destination. The cost for joining up with a club for the day is usually under 500NT, and includes transportation and guides. In my experience, the people in these clubs are all very helpful and friendly, and will go out of their way to make sure you are doing alright.