I moved to HK a couple weeks back and have done my fair share of traipsing around the city checking out the various malls and boutiques - that said, myself not being a local, take my advice for what it's worth.
By district (or rather rail stations)
1. Causeway Bay
a) SOGO -- a high-end department store. Ground floor is filled with all the large Italian, English and French design houses: Gucci, Prada, Dunhill, LV, etc, though from memory their stock was more along the lines of accessories rather than couture. Higher floors are typical department store fare. Nice if you need a new Omega watch, some bedsheets and perhaps a new tennis racquet.
b) Times Square shopping mall -- fairly immense shopping mall. Something like six floors of stores and boutiques with a "Food Forum" (three floors of fine dining, I would recommend Wasabisabi for Japanese) perched above it all. From memory: Zara, Lacoste, Fred Perry, Birkenstock, plenty of local HK brands as well. Also houses Lane Crawford, a more fashion-forward department store stocking couture as different from Sogo which is somewhat stodgier and more about homewares and accessories.
c) Kingston St surrounds -- block of streets with some nice stores, i would suggest i.t. and I.T. (yes, they're different stores). i.t. is more youth-inclined: Fred Perry, Carhartt (Euro), Recon, Swagger, etc. I.T. stocks higher-end designers: Raf Simons, Martin Margiela, D&G, etc.
2. Tsim Sha Tsui
a) Silvercord Centre - mishmash of boutiques and other youth-inclined stores. Does contain an APC store though.
b) Harbour Centre / Terminal -- huge shopping mall. Every possible designer or house, you name them - they're here. Diesel, Dita, Mont Blanc, Massimo Dotti, Zara (again),
c) Miramar Shopping Centre -- has a Uniqlo... and that's about it.
a) Pacific Place -- another large shopping mall. Another Lane Crawford, Seibu (Japanese dept store), Lanvin, LV, Ferragamo, Hermes, etc. There is a food hall in the basement which stocks fresh food and produce.
4. Mong Kok
a) Langham Hotel shopping centre -- another mall with an extensive foodcourt and quite possibly the longest escalator ever. The only store that's really leaping out from memory is a H&M and another Seibu
b) Mong Kok surrounds (Fa Yuen St) -- if you want to buy sneakers or a laptop / camera / mobile phone / pda this is the area to do it. There is a dedicated complex comprising 3-4 floors just selling computer equipment and phones, the name eludes me however a waiter in a restaurant or 7-Eleven should be able to point you in the right direction.
Apologies for the rambling and possibly incoherent post.