I assume you fly into Taipei so use the underground to get to the markets and also Taipei 101. Its pretty impressive (509m high) and the biggest food court in the known universe.
We did the train to the north and east coast via:
Su Oa - The cold mineral spring baths were great in the hot weather plus it was piped to our hotel room.
Hua lien - The national park with is jade and marble cliffs and mountain scenery is quite inspiring.
Taitung - This is the port to Green Island. This is definately worth the effort. You can hire motor scooters on the island and its rather pretty with its volcanic remnants. The best part was the hot spring by the ocean and also because the spring water is salty. (One of 4 in the world)
The steamed boiled eggs were delicious too!
We also used Taiwan as a starting point for a Japanese excursion. You can travel by "cruise ferry" from Taitung in the north to NaHa in Okinawa. Its a 20hr ferry cruise with beautiful island scenery. Flying fish and dolphins around the ship, it was teriffic. Next time we might stop at the smaller Japanese islands for more diving................... about 11 years ago
I lived in Taipei and Kaoshiung for 18 months. The north is much more sophisticated and fast-paced. But the central and south regions have more cheap food, night markets, and authentic feel to them.
Taipei 101 & Warner Village - high-end retail, movies, bookstores, etc.
XiMenDing - west side of Taipei...the cool place to hang out, pedestrian area, shops, lots of movie theatres, great street food.
ShiLin Night Market - I think this runs on Tuesdays and weekends. Its in North Taipei, look for the Mango XueHuabing -- mango snow-ice -- best thing you've ever eaten.
Chiang-Kai-Shek memorial & National Museum - huge asian looking buildings. cool grounds. not much else.
you can take the Taipei metro to all of these areas very easily.
If you have more time:
Honeymoon Bay - East coast - 2hrs by train from Taipei. Cool small surfing village. sand is not great though
Taroko Gorge - you'll need to have a car, or a mix of train, bus & taxi. Pretty national park
XiAn Mtn - highest peak in Taiwan. this is a 2-day trip. train to JiaYi, then up the mtn, or take a cab. wake up the next morning to see the 'cloud sea' with the peaks coming through.
Chaozhou - in Pingdung county, nearby is an aboriginal park with replicas, dancing etc. Chaozhou city - look for 'cold hot ice' leng-re bing -- best thing in the world!
KenTing - very southern tip of taiwan - nice beaches and night market about 11 years ago