You have three options to go from Hualien to Kaohsiung. Pick the one that makes most sense to you, I prefer flying.
- Flying: Mandarin Airlines has a daily direct flight from Hualien to Kaohsiung that takes about 1 hour. Cost is usually around $100 USD. This is in my opinion the best way to get to Kaohsiung from Hualien. All other airlines are connecting and takes over 5 hours so it's not worth it.
- Train: there is no high speed train because Hualien is on the east coast. Your only train option is to take Taiwan Railway's train 801 (regular train). Depending on the specific train, the travel time is 4:41 to 5:45 hours and costs $15-$25. Most of these trains are direct but I know at least one of them requires a connection. Best check when you buy tickets
- Driving: distance is about 350km and drive time is around 5 hours. You drive south past Taitung and then turn west to Kaohsiung
Note that there is no direct bus option available as far as I know.