1. Taipei 101. Among other things, Google Taiwan's HQ is located there. Taiwan is Google's largest hardware engineering base.
2. National Palace Museum. They hoarded all the treasures from Chinese history away from the communists and stored them there.
3. Hsinchu Science Park, the cradle of Taiwan's IT industry.
4. Shilin night market and other night markets. Food is world-famous.
5. If you go on the east coast, have some aboriginal food, like boar.
6. Catch a baseball game. They're not complete bores like MLB games are. The dancing and cheering make it an exercise event.