Taxis are a dime a dozen in Taiwanese cities. You don't need to look for a taxi - they'll be looking for you. The standard yellow cabs scour roads looking for potential riders such as lost foreigners. It is possible but generally unnecessary to phone for a taxi. To hail one, simply place your hand in front of you parallel to the ground. But they'll often stop for you if even if you're just waiting to cross the street or for a bus.
Drivers generally cannot converse in English or read Westernized addresses (except for special CKS airport taxis). Have the hotel desk or a Taiwanese friend write out your destination in Chinese, and also take a business card from the hotel. Show the driver the Chinese writing of where you are going.
As in many Asian countries, night markets are a staple of Taiwanese entertainment, shopping and eating. The most popular night market in Taipei is the Shilin Night Market (at Jiantan MRT station, NOT Shilin MRT station!), though the most exotic is arguably the Huaxi Street Night Market (a.k.a. Snake Alley) where one can dine on snake meat and try many traditional desserts. Night markets are crowded, so remember to watch out for your wallet!
Night markets are open-air markets, usually on a street or alleyway, with vendors selling all sorts of wares on every side. Many bargains can be made, and wherever prices are not displayed, haggling is expected. In the larger cities you will have a night market every night and at the same place. In smaller cities, they are only open certain nights of the week, and may move to different streets depending on the day of the week.
Bargaining is OK and expected in night markets and small stores. Computer chain shops and department stores normally have fixed prices, but at least in department stores you may get a "registered member discount" if you're shopping a lot. Anyway it's always worth a try!
When bargaining at small stores, please note that the agreed prices are normally cash prices. If you like to use a credit card, the seller normally wants to add anything up to 5% to the price as a "card fee" etc. The fee consists actually of the credit company's commission and also the local sales tax/VAT. Even if you pay cash, you normally don't get an official receipt, as then the seller would have to report & pay their taxes in full (tax evasion is rampant). If you ask for a receipt or "fa piao", you will get it but you may need to pay 2-5% more.