Ideas of things to do in Taipei?

Okay there are a few things that must be done in order to really get the full Taipei experience.
1.  Hit the Lao He Night Market on the corner of BaDe rd. and Song Shan rd. It's crowded but has some amazing food like Pork Medicine Soup, oyster omelettes, stinky fried tofu, shaved ice with fruit and BBQ sweet mountain pig sausages. There is also tons of stuff to buy and it's all quite cheap so it's easy to get carried away.
2.  Taipei 101. You've got to take a trip up the elevator and tour the super structure. We have tons of earthquakes here so the way they figured out how to deal with that was really smart.
3.  Right across the street from 101 is the Xin Yo shopping district which is full of designers brand names and restaurants. It's a really good way to spend a few hours walking around. On nice days and nights there are also a lot of street performers.
4.  Zhong Xiao east rd section 4 between YanJi st. and FuXing south rd. Is another huge shopping district. The back alleys running parallel to Zhong Xiao east rd are full of shops and interesting restaurants and tea shops. Close to SOGO Department Store is even a blood donor bus if you're in a giving mood.
5.  This will sound crazy but buy a metro card which will get you on buses, trains, the MRT (subway), and you can use it to buy stuff in almost all convenience stores. Then, rent a Ubike and cruise around the city. They just widened the sidewalks so that there is a bike lane and it's really cool to see the crazy mix of sights and sounds at ground level. It's also a tad dangerous since you need to really be careful of other drivers. It is not customary for drivers to give right of way to pedestrians. Right of way is loosely based on whoever gets there first, size of your vehicle, and the general look in the person's eyes.

I hope this helps but it only scatches the surface of this wonderful city.