In Taoyuan International Airport:

Go to the Tourist Information Desk in the lobby for assistance and information.
Transportation information: http://www.taoyuanairport.gov.tw/english/transportation/airportmap_e.jsp.

Taxis
Taxis are outside the Arrival Halls of both terminals. Taxis are available 24 hours a day. A typical taxi fare to Taipei is approx. NT$1,200 (about US$40).
Taxi information: http://www.taoyuanairport.gov.tw/english/transportation/taxi_e.jsp.

Buses
Different routes have different stops and fares.
Bus information: http://www.taoyuanairport.gov.tw/english/transportation/taipei_e.jsp

In Taipei:

Taxis
        Taxis are available 24 hours a day. They all have a yellow color and charge by the distance of travelers' trips. It is recommended that you have the name of your hotel written in Chinese on a piece of paper before you leave your hotel, as few taxi drivers speak English. In Taipei, the rate for taxi is NT$70 for the first 1.25 kilometers and NT$5 will be added to every additional 250 meters. An additional charge of NT$20 per trip from 23:00 to 6:00 next morning (depending on the time of boarding). There is no need to tip the driver.

Taipei Metro Rapid Transit System
        If you would like to experience the beauty of Taipei by yourself, the most economical and logical way for you is to use the Metro Rapid Transit System (MRT). The Taipei MRT system is well indicated with English signs. The "Taipei MRT Tourist Information and Map" which is available at each station serves as a useful guide for you to explore many interesting places in Taipei. Tickets can be purchased from ticket machines in MRT stations. The fare is based on the trip distance and starts at NT$20. Those planning frequent trips can buy a one-day ticket for unlimited travel within a day at MRT stations for NT$200. Find out more about Taipei MRT:
  http://www.trtc.com.tw/e/.   Route Map.
  http://gondola.trtc.com.tw/english.asp. Route Map.

Buses
        The bus system is reliable and efficient in Taipei. There are more than 300 bus lines and the major transfer hub is around Taipei Main Station. The fare for travel is NT$15 per section. Most bus services operate until 23:00. Find more about Taipei buses:
  http://www.5284.com.tw/TPBUS.htm or
  http://www.e-bus.taipei.gov.tw/index_6_1.html.

Outside of Taipei:

Airplane
        Taipei Songshan Airport, nucleus of Taiwan civil aviation, is located in the Songshan District of Taipei. It is easily accessible from the town centers. This airport operates on 24-hour basis with the exception of the 23:00~06:00 hour during which only limited control flights and engine test are allowed.
  http://www.tsa.gov.tw/2005tax/english/e-index.htm.
　

Taiwan High Speed Rail
http://www.thsrc.com.tw/en/index.htm