Strøget:
Strøget (best compared to Oxford Street) is probably the longest shopping street in Europe, closed for motorised traffic.
The street is connecting Central Copenhagen from east at Kongens Nytorv to west at the city town hall. Local tip is to go off Strøget and visit "Gråbrødretorv". The restaurant "Peder Oxe" is very popular amongst Danes. The square in itself is also well worth a visit.
Also just off Strøget is "Vandkunsten", "Løngangstraede", "Brolaeggerstraede" all medieval street. If you can find cafe Sorgenfri (I think BadstueStraede), they are famous for there luxurious open sandwiches as only the Danes can make them.
On Amagertorv right at the middle of Strøget is The Royal Danish Porcelain factory's exhibition which is very much addressing the tourists BUT on first floor getting close to Christmas they have an exhibition of "Royal" Christmas table settings which (even as me being a guy) are quite impressive.
Kongens Nytorv:
Is the East connecting point of Strøget. It is a big open square (Trafalgar) with Copenhagen's probably most expensive hotels (d'Angleterre). What is probably more interesting is Magasin (du Nord) which is in my opinion the best department store in Copenhagen.
Here up to Christmas you should try find a table in Hviids Vinstue, right next to Magasin. It is the oldest still existing bars in Copenhagen, and there Gluhwein (Gløgg) is to die for (well a tradition at least). This place is ludicrously busy so don't expect to get a seat anywhere.
Just east of Kongens Nytorv is
Nyhavn:
Which is the colourful house front you see on every single tourist book. During November it is too cold to sit outside, but still take a walk down and maybe have a look in Nyhavn 17, which is one of the most popular of the bars, they are all bars.
At the end of Nyhavn further East is Amalienborg, walk down to the end of Nyhavn and bear left walk along the water down towards the castle. This is where the queen lives.
If you are in the mood for more brisk walking, this will take you all the way down to "Langelinie" and the little mermaid. rather than seeing the little statue, bear left and walk up to the parks of Kastellet which is the oldest existing parts of the Copenhagen defence walls.
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At the other end of Strøget at the town hall you should pay a visit to Tivoli. At summer it is a bustling amusement park (Not like Blackpool, more like Kew Gardens with restaurants and rollercoaster's). During the winter most of the amusements are closed but there are winter stalls with Christmas decorations and what not. More important is, if you didn't get your Gløgg at Hviids Vinstue, try find a place in Tivoli to enjoy it.
The "Lake" is likely to be frozen (artificial don't worry) but it might be too early I don't know. If it is frozen this is the Copenhagen version of a ice skating ring.
If moving outside the centre:
Christianshavn has got some cute canals and "vor frelser" kirke is quite cool with its staircase on the outside of the church spire.
Don't bother about Vesterbro as this is the rougher BUT up and coming area.
Nørrebro is where a lot of students are living, thus many "funky" cafes and restaurants. "Sankt Hans Torv" is worth a short visit.
Splitting Nørrebro from Central Copenhagen are the 4 "Lakes" which are artificial but place for a nice walk.
østerbro is the nicest residential part of Copenhagen but besides from "Rosens kvarter" and "Brumlebyerne" it is quite residential.
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Traffic wise Copenhagen airport is very well connected with the town. Don't bother with taking a taxi in, as it will cost you an arm and take 3 times the time.
In the basement in the arrivals is a train that takes you to the central station in 11 minutes. Before leaving the arrivals terminal, stop at the ticket booth just before the escalators and enquire about "klippekort" (If you are planning to take buses and trains around). A blue one with 10 x 2 zones cost DKR 105.00 and each person will need
4 zones into the town centre (4 stamps).
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If you are into art, Louisiana north of Copenhagen is a very famous museum with after Danish terms a comprehensive collection of art together with short term exhibitions.
Helsingør is on the same train stretch as Louisiana museum. Helsingør is a quaint town, famous (in the world) for being were Hamlet said his famous words. (The the Swedes it is known for its liberal approach to alcohol)...
