By Taiwan High Speed Rail (THSR)
Zuoying Station in Kaohsiung is the final stop on the southern end of the THSR. There are frequent trains to cities such as Taichung (42-58 minutes, NT$790) and Taipei (90 minutes, NT$1490). From Zuoying you can then take the Kaohsiung Metro into the center of Kaohsiung, such as to the main Kaohsiung Station (11 minutes, NT$25) and Formosa Boulevard Station (13 minutes, NT$25).
By Taiwan Railways (TRA)
As well as slower local trains, there are quite frequent Limited Express services to Kaohsiung Station from cities like Tainan (35-40 minutes, NT$106) and Taichung (about 3 hours, NT$469). It’s a rather long journey on the TRA to get to Kaohsiung from Taipei (5 hours, NT$843), so most people take the Taiwan High Speed Rail.
By highway bus
Highway buses operate day and night to and from major cities like Taipei and Taichung.
By air
Kaohsiung International Airport is a short metro ride from the city center. While the number of available flights is not as large as Taoyuan airport near Taipei, it has daily connections to countries such as Japan, South Korea and Vietnam. There are also domestic flights to cities like Taipei, but as they tend to be rather expensive, most people take the Taiwan High Speed Rail.
Official tourism website: khh.travel