Taipei and Other towns worth visiting?

Danshui is right at the end of the subway line, the red line, around 40 minutes. It's right on the water with a lot of spanish history, an university, and a island you can take a ferry to to Bali. Ride a bike around Bali if you have time, but just enjoy being really close to where the river meets the ocean. It's vastly different from Taipei. Well worth the trip!

If you have the money to pay for it (it's a little expensive), the HSR (High Speed Rail) makes many towns in eastern Taiwan easily reachable for day trips. Tainan has a lot of old temples that might be of interest to travelers, particularly non-Asian ones. That's just one example. Do note that unless things have changed since I was last there that the HSR doesn't do true credit card charges for tickets. You should try to pay with cash. If you pay with a credit card, they actually do a cash advance against the card, which is terrible for you. You will accrue interest on the charge until the ENTIRE credit card balance is paid off.

One place that I would recommend that you not go to is Kaohsiung, the last stop on the HSR. Kaohsiung is a decent place in which to live, but if you don't speak Mandarin or Taiwanese, you'll get very little of traveling there. There are some museums, but little to no signs are in English in them. For short trips to Taiwan, I tell people that there are much better places to visit than Kaohsiung. If you're going to visit for a month or so then you could check it out for day or two, but for a short trip of about 1 week I wouldn't go there.

There are a number of places close to Taipei that can be easily visited as day trips, including Danshui, Jiufen and Yingge, to name just a few. Pick up either the Lonely Planet or Rough Guide books on Taiwan to look for details on these and other places.

As #2 points out, the High Speed Rail makes much of the west coast accessible as well, as long as you can afford it. Tainan, especially, is a great city to visit for its historical sights.

Lugang - HSR to Taizhong or regular rail to Zhanghua and bus to Lugang. Old Qing dynasty buildings, delicious food

Tainan - take the HSR. Free shuttle into the city - temples and more delicious food

Beipu - Guoguang bus to Zhudong and transfer, or train/HSR/but to Xinzhu and transfer - old buildings, Hakka food

Pingxi - train to Ruifang, transfer to Pingxi small gauge line - hiking and mountains, sky lanterns early next year, railroad and coal mining history, old Japanese buildings

Jiufen - gorgeous ocean views, tea, souvenir shopping, snack food - train to Ruifang, bus to Jiufen or take a taxi for set rate of NT 180

Maokong - MRT to Taipei Zoo, you can take the BR 5 bus once per hour but I suggest just taking a taxi (about 200 kuai). Good views of Taipei, good tea.

Pinglin - gorgeous mountain views on the way, good tea, small town atmosphere - bus from Xindian or Jingmei stations (it's the "Pinglin" bus, no number that I know of)

Wulai - shopping, good views, hot springs. Bus 1601 or 1602 begins from Taipei Main and goes down the entire MRT green line - you can pick it up directly outside Jingmei MRT exit 1

Taroko Gorge - take the fast train and leave really early and you can do this in a day trip

Yangming Mountain - bus 260 or Little 5 from MRT Jiantan, lots of hiking

Food & sea day trip - Walk around Bitou Cape (train to Fulong or Keelung, bus to Bitou) and return to Keelung for the Miaokou Night Market for awesome seafood, cream crab, deep fried sandwich, tian bu la etc. etc.

For details or to put together an itinerary, buy a guidebook.

Just to add to/ elaborate on/ agree with the above:

Keelung, 40 min from Taipei by train, for its night market. There are also many forts, temples, coastal scenery and other minor sights if you just want to wander.
Yangmingshan National Park- approximately an hour and a half by bus (Red 5 from Jiantan MRT sta.) from Taipei. Mt. Cising (Seven Star Mountain) is a great hike when the weather is clear- which is very unlikely. Mt. Miantian and the volcanic crater nearby (bus S6 from Beitou MRT) are also great, as is the Siaoyoukeng sulfur vent on the slopes of Mt. Cising.
I would recommend visiting Lugang over Tainan, since even by HSR getting to downtown Tainan will take a good chunk out of your day (and wallet). I would argue that Lugang is in fact more interesting- a wider variety of sights in a smaller area (incl traditional crafts shops, what I think is Taiwan's most atmospheric old street, old mansions), equally beautiful temples, also plenty of good food (though granted Tainan's food is even better).
Elephant Mountain/ Thumb Mountain- they're in Taipei, and can be walked to from 101, but they have excellent views of the city and seem somewhat ignored by most foreign tourists.

Forgot:
I would also highly recommend Sanxia's Zushi Temple- really beautiful, though a bit of a pain to get to (MRT to Yongning, bus to Sanxia Old Street). The old street is also nice.