The airport is about a half-hour to 45 minute drive from Taipei. From the airport, travelers can take taxis directly to their destination, board express airport buses or non-express city buses, or take the bus which runs to the High Speed Rail station. See location and parking map here. The following information is from the website Wikitravel (http://wikitravel.org/en/Taipei):
Express airport buses cost between NT$120 (about $4 US) and NT$150 (about $5 US), depending on the bus company, and there are stops at both terminals. Most Taipei routes are divided into West and East, with each company operating a service every ten to fifteen minutes on each route. The western line bus terminates at Taipei Main Railway Station and also makes a stop at Yuanshan MRT Station on the Xindian line (NB: The Airbus company buses on the western line meander through local towns before joining the freeway and therefore take much longer than the blue and white Guoguang buses which enter the freeway directly). Buses plying the eastern route terminate at the Taipei Grand Hyatt Hotel and make a stop at Zhongxiao-Fuxing MRT Station on the Nangang and Muzha lines. There is also a bus connecting to the domestic Songshan Airport. Ticket counters display route maps showing all stops.
In addition, there are some non-express buses which are slightly cheaper, but pass through towns such as Taoyuan (桃園), Nankan (南崁) or Kueishan (龜山) before arriving in Taipei.
When returning to the airport, express buses can be caught at various stops throughout the city. One major one is accessible via Exit 9 at the underground mall beneath Zhongxiao West Road (in front of Taipei Main Railway Station) or Exit 5 if you are coming out of the Taipei Main Station MRT. Another is at the terminal at the Songshan Domestic Airport (松山機場). Other stops are outside major hotels and also in front of Minsheng MRT Station. For people taking early morning flights, the earliest available buses to the airport leave at around 4AM from the Far Eastern Plaza Hotel (台北遠東國際大飯店) (201 Dunhua South Rd Section 2).
It is also easy to get to the High Speed Rail station from the airport. There is a bus that runs approximately every 15 minutes from the airport to the Taoyuan High Speed Rail station. From there, you can catch one of the HSR trains to Taipei Main Station (where it is easy to take a taxi or MRT to your final destination). The bus is NT$30 and the train is NT$160.
A one-way taxi fare between the airport and Taipei will cost at the minimum NT$900 (generally NT$1500-$1800 from the airport, or about $50-60 US$). In Taipei, don't make the mistake of asking a taxi driver to take you to the Taipei airport (Songshan) if you actually mean Taiwan Taoyuan Airport. The international airport is actually about an hour's drive from Taipei, while Songshan is located in downtown Taipei.
A one-way pre-arranged sedan fare between the airport and Taipei will cost at the minimum NT$1600-$2000. Generally these sedans are pre-arranged through your hotel and the sedan company or driver will meet you as soon as you exit baggage claim. Since the price is not much more than taking a taxi, it is usually recommended that you ask your hotel if they offer this service. This is a more comfortable half-hour ride to the hotel.
Direct bus connections between the airport and other cities in Taiwan are also available. U-bus also runs shuttle buses every 30 min from both terminals to THSR Taoyuan station (15 min away), from where you can continue your journey by high-speed train.