My favorite tea shop in Taipei is Wang Yo Ji. Not only does it have great oolongs, bao zhongs and oriental beauty (the 3 teas I have tasted so far), but the prices are reasonable and the store has been turned into a living museum.
In the back, the owner showed me how tea was being baked over charcoal. See the 2 pictures above. This is the traditional way. Compare it with the modern tools and you'll find similarities (the bamboo basket) but a completely different feeling. You feel you've entered a time machine and have landed hundreds of years ago! I saw it's not easy to continue using the traditional way to bake the tea. The charcoal must be evenly prepared and burn as slowly as possible. To watch the fire, one has to work even during the night!
The front of the store has been tastefully redecorated recently and features a large table to taste the tea. The second floor serves as an exhibition and conference room. Last Saturday, he invited a puppet show about Marco Polo.
A living museum, good tea, culture and a very friendly owner makes this store a must go while you're in Taipei!