Question about train/bike trip

Hello!
I am sorry that I cannot type in Chinese, but I can read. Please feel free to answer in Chinese.
I am planning a trip for next year. It will a train-bike trip around Taiwan.
We will be starting out from Taipei, take the train to one of the cities like Chiayi, bike to another city, maybe Tainan, then take the train to Taitong, and bike to Hualien. Finally, take the train from Hualien back to Taipei.
We will be renting bikes from the Giant outlets.
Any suggestions as far as the route in the west coast?
We only have about 7 days, will probably spend 3 days on the bike from Taitong to Hualien.
Thanks!