Watch out for Typhoons! thats Typhoon Season!
In additionto the last guy, try to look for the hot spring baths on the mountainside. You'll like those.
Also, don't forget to go clubbing, especially during the wedding time.
PLUSH - club located on the 12 floor of the largest "mall" in Taipei (it has a globe on one side). This club was a restaraunt by day. At night, the party gets heated. Lots of beautiful people there, and the beat keeps going till 5am.
The Basement Clubs at Warner Cinemas - When I went, there was Mega 10, a decent club near Warner Village, and another club (that I can't remember). the latter was noted to get local Taiwanese stars and celebrities, including Jolin Tsai and Jay Chou.
Carnegie Bar- located near the "Taipei Mall" (not the mall I discussed earlier)(this is more just a hotel and shopping area), its a bar with a strict age limit (even though the legal age for everything is 18): 24 for guys. but if you are 18 and female, your in. Warning: may have to dance on bar. Drunk.
ALSO, make sure (if you are a chick) to take part in hundreds of sticky photo stations around the city (little photo stickers that are like midget glamour shot sizes, except by the ATM of photo booths). chances are, you won't see those around in states, unless you live in chinatown.
Do NOT buy "Taiwan Beer" the product. Even though its domestic, its like Milwalkee's Beast threw up into a can, but not as bad as Trinidad beer (ugh). If anything, buy Kirin beer, but go for the large keg can one. Its special because it comes with a dragonhead-shaped spout you put on the top and you can drink from it directly (called kissing the dragon, by us drunk kids).
Stay away from all the franchise crap. (McD, BK, Subway, KFC). I know you wil think its like home, but you are on vacation. Sample the mom and pop shops around the city. Its cheaper and its probably more healthier. (At least get Beef Noodle soup homemade).
While shopping at the nightmarkets or shimengding, you may notice big signs with a number alongside a chinese character. That is usually a discount sign. Contrary to american "20% off!" signs, these ones actually are discounted differently. Example, IF you see a sign next to clothes or shoes that has an "8" and a chinese character, that means "price is 80% of the full price", or "20% off"
Try to bring a Taiwnese native with you when shopping. the market is hagglecity, but you get less if you are foreign, or can't speak taiwanese. Haggle aggressively, but not unfairly.
If you buy a GSM cell phonethere in Taiwan, you can use it with Cingular and Tmobile phone accounts. don't you want to be the coolest person at work!?
Taxis are life in taipei, they get you around when the buses and Subway are down. Its not expensive, especially if you have no car.
Subways are the quick way around the landmarks of the city (all the places the guy b4 announced, and me too, are accessible by subway, called the MRT, or "Jyea Ring"[don't say jyea ring]) not that expensive.
Find Bubble Tea Shops. Buy the Black Pearl Milk Tea (Boba/Bubble Tea). Drink it. Yoiu'll love it, and will fiend for it like a Starbucks.
Thats all I can think of.
If by chance, you are going to Kaoshiung instead, forget everything everyone said, because you are screwed, and that city stinks =P. Taipei is the best. Thats why its the capital! Yay.