Chiayi isn't a particularly touristy city with few attractions of interest. Tourists often just drop by en route to Alishan. The southern branch of the National Palace Museum is meant to change that. Otherwise, walk around Wenhua Road's night market and enjoy some street food, which is typical fare for Taiwanese cities.
Chiayi is 1.5 hours away from Kaohsiung or Taichung by regular train, which is within striking distance of a day trip out of either. The journey is much shorter by high-speed rail at about half an hour from either, although the station is a bit out of the city centre to make the time savings worthwhile.