The markets in Taipei, especially the night markets, are a real treat. No trip to Taipei can be complete without visiting these colorful markets. They are most lively during the weekends, and they are great places to eat, drink, shop, and people watch. The markets have everything--you can sample basic street foods, which in many instances, are the highlights of visiting a strange city, and if you are lucky, the food is so much better than what they can offer in highly priced Taiwanese restaurants. That's something to be careful about--a typical Taiwanese restaurant can set you back substantially, as they tend to offer exotic seafood priced by the Chinese but can be fairly strange, and inexplacably expensive, to anyone who did not grow up in that culture. You can buy anything there, from clothing to perfume to tacky souvenirs to kitchen tools. The most well-known, and frequently visited market is of course in the Wanhua areas near the Lungshan Temple, famed for Snake Alley.
Finally, if you are much too careless, a great place to get your pocket picked. Consider this a warning: do be more careful when traveling overseas. I have noticed that tourists, especially Americans, tend to walk the streets of most countries totally unaware, with a purse careless slung over their shoulders. This is fine in a small mid-west town, but these are sure targets in large cities like New York and Washington D.C., not to mention large Asian cities where the population density makes pickpocketing an easy feat, and careless tourists a target as easy as saying abc.