We plan to go Taiwan in Dec'12. Do we need to buy high speed train ticket in advance?

As this is our first visit there, after landing at Tao Yuan airport, we plan to go to Taizhong -> Tainan -> Kaoshiung -> Taipei. In such case, do we need to buy high speed train ticket in advance (via internet)? Or, can we buy while we are there? Is 8 days enough to cover our plan? Kindly advise. Thank you.