What's the optimal way to get from Chaiyi THSR station to Chiayi City by public transportation? According to Google Maps, there is bus 7211, but it takes almost 2 hours to travel only 15 km (I don't know how accurate this information is). Also, do buses accept Easy Card?
There is public transport available, as you mentioned. Bus route 7211 goes direct from central Chiayi and the TRA station to the HSR station in roughly 20-30 minutes - I really don't know why Google Maps is reporting this as a journey time of nearly 2 hours. There was also previously the 7212 route, but it looks like that's been withdrawn since the last time I was over.
For passengers travelling to the HSR station the journey is also free as far as I'm aware, but the last time I took the HSR was in 2014, so double-check - my subsequent journeys from Taipei have been via car. This Tripadvisor thread seems to suggest that the journey is still free for HSR ticket holders and NT$48 for those without a ticket.
Chiayi buses do accept EasyCard for short journeys. For longer journeys (i.e. to Alishan National Park) separate tickets need to be bought.
Source: My grandparents-in-law live in Chiayi.