Take Taoyuan Metro: From Airport Terminal 1 (A12) / Terminal 2 (A13) station to Taoyuan HSR station (A18).
Take THSR from HSR Taoyuan station to HSR Zuoying station.
Timetable and Fare Search: https://en.thsrc.com.tw
By taxi: It takes around 30 minutes taxi from HSR Zuoying station to Kaohsiung Exhibition Center (NT$400, ~US$14).
By Kaohsiung Metro (KRT): When you arrive at the Zuoying Station, you can take the Kaohsiung Rapid Transit (KRT) to Sanduo Shopping District Station (R8), which is then a 10-minute walk from Exit2. https://www.krtc.com.tw/eng/Guide/guide_map
By Taxi
It takes around 3.5-4 hours taxi from Taoyuan International Airport to Kaohsiung (NT$5000~6000, ~US$160~200)