hualien & sunmoon lake is not link.
go to hualien-> train from taipei down to hualien.
go to sun moon lake -> either you take bus from taipei direct or take train to taizhong and change to bus .
go to alishan -> taipei down to chaiyi stop
you can refer to taiwan map for reference.
From Sun Moon Lake > Alishan by hired cab. Cab fare: Around NT$3000~NT$4000 Travel time: About 2 hrs. Probably the fastest way to travel between these two places. It’s a much shorter route than the route via Chiayi city.
Traveling through winding mountain road can be uncomfortable to some people (motion sickness) especially children and the elderly. The mountain scenery is nice though. Some people prefer to take the more economical, longer but more comfortable route from Taroko Gorge / Hualien to Taipei city before heading southbound to Cingjing / Sun Moon Lake / Alishan.
Hualien and Taichung are separated by the Central Mountains. Take the railway train northbound to Taipei and then southbound to Taichung. For railway train travel between Hualien and Taichung, the schedule of direct trains sometime may not suit your intended departure time as there are not many direct train departures in a day. For more choices of train departure times, transfer to another train at Taipei. You can opt for HSR train ride from Taipei to Taichung.
OK! Thank you much! :)
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