I have a TR on Taiwan - http://www.fodors.com/community/asia/thursdaysds-east-asian-excursion.cfm

I didn't think much of the zoo, and you need to check that the cable car is running the day you choose.

You can take the train from Taipei to Hualien. I used a car and driver arranged by my Hualian hotel for the gorge, but there are also buses.

Have never heard of Flying Cow farm. Recommend Lonely Planet for info on transport.

I really liked Tainan, but that's because I enjoyed the temples.

There are pandas at the zoo, and the kids might like that. The cable doesn't always run, so check before you head out.

If you spend the night in Taroko Gorge, your hotel will send a car to the train station to pick you up. I'd say the train is your best bet. The gorge is really beautiful.

I've heard of Flying Cow, but never been, and don't know how to recommend you travel. Taichung is nice, and the University there has a chapel designed by I.M. Pei.

If you spend your first day at the National Palace Museum, there is an aboriginal museum almost directly across the street. You can buy a joint ticket.

There is a tour company called Edison Tours in Taipei. They might be able to provide a car and driver for you.

Have fun. Such a great country.

Skip the zoo, unless you going there to see the pandas they have specifically.

You can take a train from Taipei to Hualian, go to Taipei Main Station to buy the train tickets. I use to take airplanes to Hualian, but I am not sure if they still have domestic flights in Taiwan (since they have high speed rail now).

I personally like Ximending Market area in Taipei, Shilin Market has been too touristy for me and the food quality has gone down, but still very interesting for first timers.

The science museum in Taichung has a section about Chinese technology from the old times, that is the only exhibit that really caught my eyes.

Taichung has 3 night markets, with Fengjia as the biggest one.

Other places in Taichung that are interesting: 1. Taichung Park 2. Taichung Confucius Temple (near Taichung Number One High School, which also has a small night market in front of the school at night) 3. Wen Chang Temple.

oh, for CingJing, here is their English website with activity suggestions: http://www.cingjing.gov.tw/en/trip/index.php

When I was there, I went to watch the herding and farm related shows. They let you feed the sheep milk.