I recently brought some friends over for the first time:
- day one shopping for the ladies and eating - TPE is a place to experience street life, shop for cute stuff, rather than serious retail therapy.
They have a lot of small shops for you to check out, see and try cute things. Tea is also nice, look for Taiwanese Longjin Tea. Good on the gut too.
For men, the hifi scene is nice if you like Taiwan stuff, like Usher. Otherwise foreign imports are cheaper back home. Plus electronics are 110V.
- day two
Hire a taxi driver, and get him to bring you out. A day out to Jiu Fen and Shi Fen (九份 & 十份）- a must for first timer
- day three
Explore town - the 故宫博物院 museum and the Taipei 101 are worth visiting. See the change of change at the Chiang Kai Shek Memorial
- Day Four
Yamgmingshan 阳明山 is nice if you have time