Taipei City Tea Houses:
In the area around #2 and #3, there are many tea shops to visit. Walk around the small streets and float by anything that seems interesting to you. You will find some fun tea or teaware stores to visit. Also, this area is home to the original soup dumpling restaurant (Ding Tai Fung). I would definitely recommend visiting early. Go before 10am to avoid the crowds and enjoy breakfast.
Make sure to go for the traditional Taiwanese food, and not Japanese food. My favorite dish is baby oyster with black bean sauce & the rice crepe roll)
Little towns outside of Taipei:
Sun Moon Lake:
The most beautiful hotel by the lake
(Make sure to take in an early morning walk for full effect)
Taroko gorge National Park: