By Taiwan High Speed Rail (THSR)
All THSR trains stop at Taipei Station and Nangang Station in the city center. There are frequent trains to cities such as Taichung (50-60 minutes, NT$700) and Zuoying in Kaohsiung (1 and a half to 2 hours, NT$1490).
By Taiwan Railway (TRA)
While there are a few train stations run by Taiwan Railways (TRA) in the city, you are unlikely to use them everyday unless your accommodation happens to be on a TRA station outside central Taipei. Some tourist destinations outside the city can be accessed via TRA though, and for these you can first easily get to Taipei or Songshan stations via the metro, then join a TRA train.
There are frequent services, but check the official TRA website for longer journeys outside the city as express train services are often sold out. There are express trains to cities such as Taichung (100 minutes-2 hours, $375). While not as comfortable as the high speed trains, these express services are cheaper and take you right into the city center.
By highway bus
Bus services are provided day and night to and from the city to all over Taiwan. The main Taipei Bus Station is located near to the main station, but there are also services from/to other spots in the city, such as the bus station near Taipei 101.
By air
The city is served by Taiwan Taoyuan International Airport, a modern airport outside the city, and Songshan Airport, located in the city center. Most international services operate from Taoyuan, from where you can take the Taoyuan Metro into the city (39 minutes on the express service, NT$150).