Taipei's MRT system can take you to many interesting places that all offer something different. You can pick up a free English map at any MRT station that will not only help you find your way around the city, but will also give plenty of ideas about other places to visit.
If you are hungry you can visit one of Taipei's night markets. There are many museums and galleries. If you want a good view of Taipei then you can go up Taipei 101, ride the Miramar Ferris Wheel or the Maokong Gondola. If you don't want to pay for a view of the city then it is easy to hike up Elephant Mountain or other peaks around the city.