Additional notes.

The AirBus costs $3 between CKS and Taipei; the coaches are air conditioned, clean, and comfy enough. There are two routes, one going to the downtown hotels, the other to the ones to the east.

I'd say that your number one museum attraction in Asia is just outside Taipei - the National Palace Museum. Taxi ride costs maybe I dunno $15, and you can stay all day and then take one of the numerous city buses back (they leave from the main drag below the museum).