Strøget is the main shopping street in Copenhagen. It's a pedestrian street, and the real names of the streets which constitute Strøget are (starting from Kongens Nytorv and moving towards Rådhuspladsen) Østergade, Amagertorv, Vimmelskaftet, Nygade, Gammeltorv, Nytorv and Frederiksberggade.
This is where all the tourists- but also the Copenhageners- can be found. This is where you find all the great shops, the very expensive ones, the medium priced but also the cheap high street chains such as H&M, Zara, Top Shop, Mango, Cos etc. The area is also filled with cafes. Try to avoid the ones placed directly on Strøget, since they are the most expensive ones and often a bit touristic (exceptions are Cafe Europa and Cafe Norden, which I really like although they do have higher prices than most cafes in Copenhagen). Instead you should visit some of the many cafes located in the smaller streets behind Strøget. Especially the so called Stræderne (Læderstræde etc.) are full of great cafes and small shops.
Also the small streets which you find in the area between the two shopping centers Illums and Magasin are full of fantastic shops. Start "behind" Illum in Kronprinsensgade and continue straight on to Store Regnegade and Ny Østergade. Also Gothersgade has got several great shops in the end near Kongens Nytorv.
Don't forget to visit Gråbrødre Torv which, with it's colorful houses, is probably the most beautiful and charming, yet quite "touristic" square in Copenhagen. On a warm summer night this is the perfect place for an outdoor dinner.