Seeking advice for my Taiwan trip

Hi I am KW, a senior citizen from Singapore. 4 of us will be going there for a 2 week-free & easy trip from 3/11 to 16/11. We have so far planned and booked for acommodation for a 2-day trip in Taipei followed by a 2-day trip in Cing Jing farm. Then we intend to visit Alishan, Sun moon lake, Rukang, Koasong, Hualian, Taroko Gorge, Yilan and back to Taipei to cover more places of interest. Can you please advise us how to go about leisurely by means of public transport such as bus, railway, bullet train etc. Is temperature in Alishan very low? Do we have time to visit the Jade mountain?