National Palace Museum

Good place to visit when you are in Taiwan. Lots of collections of arts and exhibits. There's also a garden by the entrance of the museum. You should go after the museum. You can get in for free (just show your ticket stub) Plan to stay there about good 3-4 hrs.
Bad thing about this place is you can't take any photos inside, even if it's w/o flash.

This is the best museum in Taiwan containing real treasures from
China's long history. A lot of the pieces here are from the imperial
collection, so it would be hard to get any better! You will find
bronze, jade carvings, china, ivory carvings and sculptures, curio
boxes, paintings, calligraphy in special or permanent exhibitions. For
example, a special exhibit was the "imperial delights: prized
possessions from the Ching Imperial collection". One of the most famous
jade carving was a white and green piece of jade, carved to look like a
Napa cabbage, using the green and white part of the jade to the best
effect, and there was even a praying mantis on the cabbage. There are
also buddism books which were hand written using gold ink on dark blue
paper, written up to 1000 years ago, they were so well preserved they
look as if they were written yesterday. Plan at least half a day at the
museum, or if you have the time, then 1 or more days will also be well
spent here. Outside the museum itself, there are 2 Chinese style
gardens, Chih-shan Garden and Chih-te Garden, both marked on the museum
information guide map. The ticket to the museum was 100 NT (about $3).
Free scheduled docent tours are available in many languages.