A spring board for the rest of Taiwan.
Taipei is wonderful, but don't neglect the rest of the island. Once you get your fill of night markets, temples, and museums head out of the city. You can get into the mountains relatively easily with bus service, but some spectacular locations can only be reached with a car or scooter. Hitchhiking works, but I recommend that one does so with a friend, a smile, and some decent conversational Chinese.