There are two international airports in Taiwan:

Taiwan Taoyuan International Airport (TPE)

The Taipei TTY Airport is the main gateway of Taiwan. It can be accessed via over 190 non-stop and direct flights from 56 major cities throughout the world.
Official Website

Taipei Songshan Airport (TSA)

Situated in Metropolitan Taipei, the Taipei International Airport is the hub of domestic air transportation. It is easily accessible from the town centers.
Official Website

Transport between Taiwan Taoyuan (TTY) International Airport and Taipei

Taxis queue outside the Arrival Halls of both terminals. Airport Taxis charge according to the meter plus a 50% surcharge (highway tolls not included). Typical fare to Taipei is around NT$1,100.
Four bus companies now provide frequent service between TTY International Airport and important destinations around Taipei everyday in every 30 to 40 minutes. Costs are approximately NT$110~140 (US$4~5), takes about 60 to 90 minutes. The bus station is on the southwest side of the Arrival Passenger Reception Area in Terminal I, and on the northeast side of the Arrival Passenger Reception Area on the first floor in Terminal II. For more information, please check the website: Official Website

Taoyuan International Airport MRT桃園機場捷運

Connecting downtown Taipei City to the Taoyuan International Airport (TPE), the Airport MRT is the fastest way to reach the airport, with express service taking 35 minutes from Taipei Main Station to both Terminal 1 and Terminal 2.
Two types of train service are available for riders: express service from Taipei Main Station to the airport takes 35 minutes, and commuter service stops at all stations. Express and commuter trains alternate with a 5 minute interval until midnight. Express trains are colored purple, and commuter trains are colored blue.
Passengers flying on participating airlines can utilize in-town check in and baggage services at Taipei Main Station.
The ticket fare from Taipei to the airport is NT$160, and may be paid by cash at the ticket machine or booth, or by Easycard (悠遊卡) or iPass. Passengers riding between A12-A14a stations within the airport grounds are free if using an Easycard/iPass. Commuter passes are available for 30, 60, and 90 day durations.
The first express train leaving Taipei is at 6:00AM, and the last express train leaving Taipei is at 22:58.
The first express train leaving the airport to Taipei is at 6:12AM, and the last express train leaving the airport to Taipei is at 22:42.
The airport MRT line stops at the following stations:

Official Website

Transport in Taipei

The system operates according to a spoke-hub distribution paradigm, with six lines running radically outward from central Taipei. The metro system operates from 6 am to midnight daily with extended services during special events. Trains operate at intervals of 1.5 to 15 minutes depending on the line and time of day. A “Single-journey Ticket” costs between NT$20 to NT$65 depends on distance. A NT$200 One-day Pass purchased from a service booth will allow unlimited travel among all MRT lines within one day. Smoking, eating, drinking or gum chewing are strictly prohibited on all MRT trains.
For more information please visit: Metro Taipei

Mass Rapid Transit System (MRTS)