In addition to your thinking about ideas in the threads cited above, you will probably want to spend most of your time in Taipei, but with 6 days you should also get out of the city.

The MRT makes Taipei easy to get around. Taxis are also easy and fairly cheap in Taiwan, though you will need addresses written in Chinese. (Also, most drivers won't take 5 people in one cab.)

I would suggest an overnight trip to Tainan via high-speed rail. And definitely an excursion to Taroko Gorge, possibly a one-day bus trip.