Personally I recommend using an app to get taxi in Kaohsiung. Though prices are very similar to traditional taxis, it's much more convenient. The best apps are
Uber: same as elsewhere in the world
FindTaxi: very good app that lets you pick what kind of cars you want (like female driver, booster seat, accepts credit card, non smoking, and so on). You can get if for both Android and iOS
You can also call traditional taxis. You can either hail one on the street, call their dispatch number, or go into a nearby convenience store or hotel to call.
Here's a list of all the registered taxi companies (as of 2019) serving Kaohsiung and their dispatch phone numbers. Some of them also have apps, check their websites. The ones near the top are the bigger ones: