Domi - Taipei has some of my favorite cafÃ©s in the world. Make him give you the names of his favorites instead of coffee! ha ha.
Keelung (north port city) or Kaohsiung (south-central), and travel in between. At least one time, drive down the east coast - very scenic in places. Rocky cliffs at times, sandy beaches others.
The best food per mile in the world.
Oh. And get a driver - don't think about driving yourself. Rome driving is EASY by comparison!
EDIT: actually, there is a market in Keelung - kind of a big, sprawling open place that has THE BEST seafood in the world. Truly "right off the boat". Point to some fish, point to the kettle or wok, and enjoy sauces and taste treats.