And if you have trouble call the 24-hour tourist hotline: 0800-011-765. They speak English, Japanese and Chinese and can answer pretty much any question and will also help you to talk to a local, such as at a hotel if needed.
If you are under 30 you can pick up a free cell-phone (you have to return it when you leave) at the airport when you arrive. It’s got a built in English-Chinese dictionary.
Also, there are good tourist offices in the train stations of all the major cities. Staff speak English. So Taipei, Keelung, Tainan, Kaohsiung, Chiayi, Taidong, Hualien, etc.
You don’t need to spend any time in Taidong and Hualien. They are ugly little towns. Lots of cool places to stay between these two. Sun Moon Lake is pretty but it’s really more of a local attraction for all the promotion it gets. If you are here by yourself, on a short trip, you can miss it. On the other hand, it makes a good base if you want to go to Puli (and the Snake Kiln in Shueili).
Don’t forget to take the new gondola up to the hills of Maokong when in Taipei.
Puli: I take it you are going for the Chung Tai Chan Temple. Make sure you call or email first to book a tour. Occassionally they have a day off.
Alishan. There is a great new hiking path linking Fenchihu and Rueili. Get off the train at Fenchihu and head up the main road. The trailhead is obvious and there are English map boards the whole way. Lovely walk through bamboo forests. About 4-5 hours.
Your itinerary is a bit skewed. From Taroko you would head south to Hualien and Taidong. From Taidong take a train to Kaohsiung and then work your way up to Tainan, then Chiayi (the start of any journey to Alishan), then Taichung for a bus to Puli or Sun Moon Lake.
Keelung and Beitou are day trips from Taipei.