My favourite local place is the area around Elmegade, where different shops and cafés are located. No market though.
Elmegade is a really nice area! There you'll find really cool shops and cafés (try Laundromat, for instance).
Elmegade is a great small street with some special stores. Strøget though, has the widest range of shops, but is often crowded.
For shopping in and around Vesterbro: check out http://preview.tinyurl.com/ck5ch4