LITTLE CITY-GUIDE:

KOPENHAGEN CENTER
Torvehallerne - get off at Norreport Station and you'll immediatly stumble across a large hall whereyou will finds lots of cute food stalls, organic cosmetics & other things-great to have breakfast or refresh with a freshly pressed juice, there is also a produce market in outside which you can see in one of the pictures
King's Garden - large Park, ideal to soak up a little bit of sun on a chilly day or have a little Picknic under one of the trees- except for one park area it wasn't really crowded when we went
Rundetårn (circular tower)- the only tourist attraction we spend money on, for only 3€ you have a spectular 
view all over the city  but be prepared-there is no elevator!
Gråbrødretorv - a small square where the times seems to stand still, very relaxing atmosphere where you can take a break from the crowded streets around
Wasteland Vintage/Studiestræde 19 - rather large Vintage Store selling tons of second hand items for reasonable prices

NØRREBRO

Jægersborggade - very cool&hip street with many small stores/boutiques selling all 

OTHER
Amager beach park - even though this isn't the most natural beach you'll ever see it is still a nice place for a getaway from the city's rush-nothing better than laying down in the sand and listening to the sound of the sea, especially if you don't have much time to visit a real beach that is further away
Rent a bike - the moment you arrive in Copenhagen you will immediatly see that this really is the city of bikes. The bike lanes are enormous and they even have their own traffic lights. Rentig the official city bikes with a build in gps does only cost around 3€ an hour and you can explore the city from a whole new perspective. We rode our bikes from the city center to the beach for 40 min instead of taking the bus and it was 100% worth it. You can leave the bike at almost every metro station if you ever get tired and decide to continue by public transport

Also check out my "What I Ate in Copenhagen" video for some tips where to eat.