1. Taipei 101 building, mall, etc. in the Shinyi district.
2. National palace museum.
3. If you like to have deep dive of cultural/religions aspect of things, take MRT and get off at Yuanshan station, stop by Buddha's temple (104, Taiwan, Taipei City, Zhongshan District, Yumen Street, 9號), then Taipei Confucius temple (Confucius), and lastly Baoan temple (Daoism). All these three places are within short walking distance from MRT station.
http://zh.wikipedia.org/wiki/%E8%87%A8%E6%BF%9F%E8%AD%B7%E5%9C%8B%E7%A6%AA%E5%AF%BA
4. Yeliou area at Northern sea shore, or Beitou hot spring.
5. Tamsui. Take MRT Tamsui line all the way up to the North. There are a lot of things to do in Tamsui, the old street, the fisherman's walf, old Spanish fort (Fort Santo Domingo), etc.