In Taipei, go to the area that's the center of the textiles trade. Get to the corner of Nanjing West Road and Di Hua Street and walk north about a block or so. There is a large building with several floors of textiles fabric vendors and seamstresses across from 7-11. If you want to choose fabric for tailored clothing, this is the area to go. Numerous shops nearby also sell fabric, plus all the extras that you'd need. Buttons, zippers, clasps, beads - everything. You'll also find many dress shops and tailor shops on the streets nearby. My wife and some of her friends had some dresses made for them at a dressmaker there. If you don't speak Chinese, you might want to take a friend who does.
You might also try the shopping arcade on the corner of Section 3 Zhongshan North Road and Nong An street, by Florida Bakery. The shopping arcade houses a lot of Filipino shops on the second floor. Several Filipina seamstresses have set up shop in the arcade, and as they speak English, this might be a good place to check out.
Finally, seamstresses can be found in several of the public buildings housing Chinese markets in Taipei. For example, the Si Ping Street market (Songjiang Road)has seamstresses nestled in the interior of the building, while the butchers and vegetable vendors inhabit the walkways nearer the streets.
You could also ask a friend or neighbor if there is a seamstress near your house. In my experience, all neighborhoods in Taipei have seamstresses, with signs that read "alterations" in Chinese. I can read the characters, but I can't type them. Sorry.
As for prices, the seamstress near my house in Sanchong charges $150 NT for taking in a men's shirt. I'd consider this the low end, as this is not an upscale neighborhood. I'd imagine the work you want done in Taipei may cost more.