I did a trip report last year on my first time trip.
http://www.flyertalk.com/forum/trip-...tg-c-ua-f.html

I think the National Palace Museum is a must.

101 has some interesting exhibits on the observation deck level. Not cheap but I thought it was worth doing. Would be even more worthwhile on a clear day.

I think Elephant Mountain is worth the hike if you are up for a whole lot of stairs.

Use the MRT, it's fantastic! I'm a bit of a public transit buff so maybe it was more exciting to me, but I loved it. Clean, bright, extremely cheap.