I recently spent a week in Taipei.
It's hard for me to put my finger on whether it was too much time or not, but I felt that a week was enough time for me to happily appreciate and understand the city intimately. I mainly spent my time in Taipei, and went out to smaller surrounding villages/towns on day trips taking the train and whatnot. I posted some photos of my trip on my
blog, if that helps. I feel that it's best for you to figure out what your objective is, and see how you can achieve that goal before or while you are in Taiwan.
If you are looking for a mobile SIM with a data plan, like DragonChi said, the moment you exit the doors after Immigration and into the arrivals hall, make a left towards the bakery/cafe, and restaurant. Immediately to your left, you'll see several mobile companies lined up there catering to tourists. You will need your passport to register for the service. My understanding is that locals are not eligible for the plans offered at the airport
(I may be wrong, however.)
You can apply for the tourist SIM in advance to expedite the application process for several of them and to make your life easier. For reference, I used
Chunghwa Telecom. Feel free to shop around to find what caters best to your needs; a quick Google search will yield some good results. Also, if you are spending a week there, you can buy a tourist-oriented MRT card for the subway/bus at the arrival hall, located in the middle'ish of the arrival hall (you can't miss it.) Interestingly enough, I found that a lot of restaurants/shops still take cash only, and will post signs they do not accept credit cards.
In terms of where you must go, again, that depends on what you are looking for. I went to Taipei really as a pre-cursor to my China trip (which I am dreading.) As a handful of friends said to me, Taiwan is similar to China, except it's cleaner, and people are much more polite and rule-abiding (i.e. no jay walking, lining up properly etc.) And they were right about that. Public health is highly emphasized. Washrooms, no matter where you go, are cleaned regularly, even at the subways. There isn't much garbage/litter floating around on the streets. I suggest bringing a few packs of Kleenex because some places will not provide them to you.
For what it's worth, I went to the Taipei Zoo, took the tram up to Maokong Mountain to try the teas (not much action up there.) Tried to go to Taipei 101, but couldn't go up because it was already too late. We also took the train out to Jiufen and Shifen village, and the Huotong Cat Village (be forewarned - no taxis come here, so check the train schedule and don't be late!!!) Most of my time was spent exploring the back alleys and the coffee culture in Taipei (Fika Fika is highly recommended.)
All in all, I liked Taiwan, but it would be hard pressed to spend another week there, unless I am able to re-connect with my family there (which did not happen as they were away on business.) Most places felt like, and were exactly tourist traps, but it is what it is. The people, I felt were very genuine and friendly for the most part. The language barrier was probably the biggest of my struggles. As for "jiu dian" - I did not know about them until I read this thread. I will admit, some Taiwanese girls are smoking hot. Let's just say, I got in trouble a few times and got slapped on the wrist by my female travel companion.
Let us know how it goes!