SOGO: located in Zhongxiao Fuxing station (blue line).
photo from gilron.netThere are several Sogo department stores in Taipei, but I am most familiar with these two. The white building is Zhongxiao Sogo, which is one of the earlier Sogos, and also the one that I grew up going to. Every hour, the clock in the from plays "It's a Small World." Worth seeing once, I think. There's a Junkudo in this Sogo, which, I think, is the best place to buy Japanese books in Taiwan. Manga section is HUGE. There's maybe 12 floors in this Sogo that carries things ranging from cosmetics to children's toys.
photo from soeperbaby on flickrThe blue building is the Fuxing Sogo. It's pretty much just right across the street from Zhongxiao Sogo. I've only been to this Sogo once, but it carries all the designer brands. Some that I can remember from the top of my head are Burberry, Chanel, McQ, Vivienne Westwood, Mulberry. And then there's a lower floor that has Nine West and Aldo. That kind of stuff.
Photo from crackingtheegg.wordpress.com
photo from wikipedia
Hankyu Department Store (阪急): Located at Taipei City Hall Station (blue line)
photo from fastretailing.com
Taipei Underground Metro Mall: located in Taipei Main Station (the big station; red/blue line)
photo from panoramio.com
photo taken by me
The first picture on top is Zhongshan Metro Mall. The bottom two are from Taipei City Mall. There's a place on the left side all the way near the back in the Taipei City Mall that sells ice cream for a good deal. It's a little more than a dollar for two big scoops. Sometimes I wish we had these deals in America. We make up for it for having affordable Haagan Daz. A small pint sized one costs about $10 in Taiwan.
Eslite 誠品: located closest to Taipei City Hall station, next to Shin Kong Mitsukoshi (blue line)
photo from Wikipedia
Photo by mom
photo by me
Taipei 101: located closest to Taipei City Hall Station (blue line)
photo by me
photo by mom
Ximending: located on Ximen station (blue line)
photo taken by mom
photo by heart of serpendipity
Shilin Night Market: get off at Jiantan Station (red line)
photo by stan2china
Danshui/Tamshui (淡水): located on Danshui station (red line)
photo by nihaonickandleah.blogspot.com
From Danshui, you can take a boat to Bali or the fisherman's wharf.
photo by mom
photo from gilron
Anddd there are the big places that I suggest! Shopping in Taiwan is fun, and there are shops everywhere in Taipei. Even stores like 7-11 and Watsons (convenience/ drug store) are fun. If you're looking for something more specific, I can't promise I know where to find what you're looking for, but feel free to shoot me a question.
Department stores carry a lot of Japanese imported brands and higher end Taiwanese brands, as well as imported foreign ones. Some of my personal favorite Taiwanese clothing brands are Gozo, Art Icon, a la sha, and J Daheck. Street brands are always nice, though, because you can look through pieces that have been picked out already and then haggle.
Taiwan is still constructing and rebuilding, so no matter what, this will still be incomplete.