Taipei to Taichung and Kaohsiung

I would like to travel free and easy from Taipei, Taichung and Kaohsiung between 11 to 20 Oct 14.
May I know what is the best way for my travel plan to connect my train from Taipei to Taichung and then Kaohsiung. I plan to arrive in Taipei on 11 Oct 14 at 4.40 pm at the airport and leaving from Taipei Airport on 20 Oct 14 at 1.10 pm. Should I go directly from Taipei to Taichung first for 3 nights and then to Kaohsiung for another 3 nights and last three nights stay in Taipei. Most likely I will take a HSR from Kaohsiung to Taipei. But from Taipei to Taichung what is the best transportation?
Your suggestion pls. Thank you so much.
Hill