Taipei is a modern Chinese city that is becoming more modern and more Americanized every time I go there. It's not a city for remarkable sights or hidden luxuries, but it's quite easy to immerse yourself into the daily life of this city.
There aren't many foreign tourists visiting Taipei, so it is easy to explore the local sights without being "touristy."
The things one must do there are visit the national museum, eat as many different types of fruits and foods as possible, and explore the city by foot and bus and subway. You'll get a very good flavor for daily life in a modern Chinese city.
Quick Tips:
When traveling from New York City, I highly recommend Evergreen Airways. The flight is comfortable with excellent service and goes through Seattle. About 20 hours getting to Taipei from Newark.
Best Way To Get Around:
Taxis are cheap compared to New York City and plentiful. An average ride of 5 miles or so costs about 60 NT, or .00. The new mass transit railway system is quite easy to use and also very cheap, with the most expensive fare costing about US .00.