Def recommend the the National Palace Museum...not many places in the world where you can view 7000 year old artifacts from the same civilisation. The pieces in that museum are stunning and they say if they rotate the viewing collection every 3 months, it would take 15 years or so to see it all.
Bars and pubs to visit are The Pig, Sean's at the Westin and of course, Carnegies.
Other than this, food is excellent! Def recommend the night street markets...watch out for chou dofu! (stinky tofu)