What is the best way to travel either from Taipei or Taichung to Alishan?

You can take a train to Jiayi first and then change to tourist bus to Alishan. From Taipei, the high speed rides will take around 1 or 1.5 hours to Jiayi high speed station. The cost would be around USD35 per person. Then you can take the tourist bus to Alishan from there. The tourist bus may need around 2.5 hours. You can also take normal speed ride to Jiayi, the duration would be around 3.5 to 5 hours. The tourist buses from Jiayi Normal Speed railway station to Alishan also take around 2.5 hours.

High Speed rides from Taichung to Jiayi would be around 30min or so, the price should be around USD14. The normal speed ride would take around 1 to 1.5 hours.