How to get there
It is possible to get to both Alishan Forest Recreation Area (Alishan/Zhongzheng Village) and Fenqihu by public transport. But beyond that, if you don’t have your own transport you will be severely limited in terms of what you can do. Hiring a scooter at Chiayi is a good option. There are rental places at the HSR station.
High Speed Rail/Train
You can take the train/HSR $598/1080NT from Taipei to Chiayi and then transfer to a bus to make your way up to Alishan. Buses leave every 30 mins or so from Chiayi to Alishan Village. Tickets cost $221 each way. There are two buses each day that leave from Chiayi to Fenqihu. Tickets are $161NT each way.