Well, Kaoshiung is at the south end of Taiwan Island, so the suggested itineraray is Taipei->Taichung->Kaoshiung->Hualien->Taipei. This is just a circle-trip round the Taiwan Island. All of th journey could be finished by train. Express train from Taipei to Taichung willt take you about one hour, and then you could take a bus or train to Kaoshiung quickly. Finally, you could take train back to Taipeo from Hualien in 2 hours.

Taipei-Kaoshiung: around 2 hours by train.
Kaishiung - Hua Lien: 6 hours by train
Hua Lien - Thai Chung: 5-6 hours
Tai Chung - Taipei:around 1 hour