From Taiyuan Airport to Taichung, you can take high speed train, it takes around 40 minutes to arrive and costs about NT$480-540. From Taichung to Chaiyi, you can still take high speed train, it takes around 25 minutes to arrive and the ticket costs NT$340-380. From Chaiyi to Alishan, you can take train or bus, I think you can take a stopover at Fenchihu, this scenic area is worth visiting. YOu can take train or bus from Chiayi to Fenchihu and then take train or bus from Fenchihu to Alishan. Here is a phone number of local guide with car renting service since recently the road to Alisha has been affected by the flood, if there is no train or bus to Alishan, you can call this number: 0985276999 . From Alishan to Taitung, you can hire a van to finish the trip. By the way, why don't you go to Taroko, this is the best place for backpacking. From Hualien to Jiufen, you can take bus to Keelung，and then take bus to Jiufen, you can tour around the night market in Keelung. From Jiufen to Taipei, take bus first to Keelung and then train or bus to Taipei.

Thanks for your help especially the Alishan journey and your recommendation for the Taroko. Any info for the sharing van hire from ALishan to TAitung section. We are two girls; is it safe?