From Taipei to Beitou, take MRT to Beitou Station, from Beitou to Sun Moon Lake, take MRT back to Taipei and then take high speed train to Taichung and then take train to Sun Moon Lake. From Sun Moon Lake to Alishan, take bus to Taichung, then take bus to Chiayi, and finally take bus to Alishan, if you found the route is much complex, you can hire a van and finish your trip. From Alishan to Taoyuan Airport, take bus to Chiayi and then take Ubus to Taoyuan Airport.