There is no direct service from Taichung to Green Island. Either go south to Kaohsiung and take the South-Link Line to Taitung, or take HSR to Zuoying and transfer to a Puyuma Express / EMU3000 Tze-Chiang. All-TRA end-to-end is about 5 hours; HSR-plus-TRA is about 3–3.5 hours — faster, but a notable jump in fare.
Once in Taitung, board Puyuma Transport Route 201 at Bay 1 to Fugang Ferry Pier (Taitung) — about 20 minutes, NT$25 — then take the jointly operated Kaixuan / Tianwangxing / Lyudao Star boats to Nanliao Harbor (Green Island), about 50 minutes.
Green Island runs 2 fixed sailings on weekdays (09:30 and 13:30); in high season and on weekends and holidays it scales to 5 sailings a day (07:30 / 09:30 / 11:30 / 13:30 / 15:30).
A 06:00-ish HSR departure from Taichung puts you in Taitung around 09:30, in time to catch the 11:30 bus and connect to the 13:30 sailing. All-TRA via the South-Link Line gets in after 13:00; the next bus from there is 15:10, arriving Fugang at 15:30 with a zero-buffer connection to the high-season last sailing — and on weekdays there is no boat in that window.
If you cannot connect the same day, overnight in Taitung — stays around Taitung Train Station are plentiful — and aim for the 09:30 first sailing the next morning. With an extra day in your schedule, spend a night in central Taitung or somewhere along the South-Link Line before boarding.
The South-Link Line is often suspended during typhoons or heavy rain (July–November); check TRA announcements before you leave. Buy ferry tickets and confirm capacity on ferry.tw.