You can also take the MRT to the end of the Tamshui line and have a nice stroll along the water. Kind of honky tonk, but there are some pleasant diversions. The college at the top of the hill is a nice lookout point. I enjoy hiking along Yangminshan mountain. There are some hot baths that are worthwhile. If you miss westerners, eat at the American Club Sunday morning buffet (you may need a sponsor.)

Chiang Kai Shek memorial is worthwhile, imho, and the Handicraft Mart just across the street is a good place for souveniers and jade. You can walk all around Taipei and hit the provincial history museum grounds, CKS memorial and the concert hall/theatre buildings, presidential palace building, and history museum/botanical gardens all in a couple hours. None of these museums approach the national palace museum, but to get a taste, they are good.

if you get out of town, Chiu Fen (not sure on spelling) is a neat town built into the side of a hill. Yeliu is a great place on the coast to see some interesting rock formations. Keelung has some gun fortresses from WWII. Again, there are lots of hills and mountains to hike. Take tea at a tea house. If you have lots of time, head out to Hualien and Taroko gorge.