Concierge.com's insider take:
Kolonaki is the major fashion shopping district, pleasantly packed with local and unusual designers as well as the typical international houses of luxury. Plateia Kolonakiou is the center of it all, plus Voukourestiou and Skoufa streets.
Off Syntagma and stretching all the way to Monastiraki, pedestrianized Ermou Street is the more egalitarian shopping mecca. Chain stores abound, but they're interspersed with more typically Greek, interesting stores.
Monastiraki hosts a vast and jammed flea market every Sunday; it snakes around side streets and stretches west to the most flea-heavy section in Plateia Avissinias.