Copenhagen & Copenhagen Kastrup Airport (CPH)
Copenhagen Airport: Called Kastrup, the airport is very close to the city center. It is easiest to take a train to København H, which is the central station in Copenhagen. It is only 3 stops from the airport and takes about 15 minutes. As of May 2010, the ticket cost 34,50 DKK. You can buy them from the machines if you have enough coins or a card with a chip (they won't accept regular American credit cards) - simply choose Ticket and København H as the destination. The ticket counters are right next to the machines. You won't need to punch this ticket when you get down to the tracks because those machines are only for clip cards. The central station stop is announced in English, so it's hard to miss! Alternatively, you can take the Metro to Nørreport, then change to an S-train to København H. You will need to buy a zone ticket for 3 zones, but is it the same price as the train.
The public transportation system in Copenhagen is extremely easy to use. It's usually ranked the best in the world, so try it out! You can use the site Rejseplanen to see which trains to take.
If you exit central station through the main doors instead of the side doors, you will be looking directly at Vesterbrogade. You will see hotels on both sides and a small road for drop-offs. If you go straight out to Vesterbrogade, and turn right, there is an information center on the corner where you can get a free map of Copenhagen. If you continue past the information center, you will come across Tivoli Gardens and eventually make it to the City Hall and pedestrian roads.
Museums: A lot of the museums have free admission, such as the National Museum and Museum of Danish Resistance 1940-45. Some charge between 40-60 DKK for admission, and Ny Carlsberg Glyptotek is 60 DKK, except free on Sundays. Most museums are closed on Mondays.
Most of the main tourist sites are within walking distance from the downtown area. The Little Mermaid statue is a little farther out of the center, so you may want to take an S-train to Østerport station. Unfortunately, the statue is currently in China for the World Expo and will not be back in Copenhagen before the end of 2010. There is an original copy of the statue in the pond at Tivoli Gardens, but you have to pay 95 DKK to enter.
There are three main palaces/castles in Copenhagen: Amalienborg (where the royal family lives) for 60 DKK, Rosenborg for 75 DKK, and Christiansborg for 70 DKK. The line at Amalienborg is rather long, so make sure you have plenty of time to wait in line. The crown jewels of the royal family (15-19th centuries) are in the treasury at Rosenborg.
Canal Tours are an easy way to see a lot of the sites along the water. Guided tours in Danish, English, and either German, Italian or Spanish (depending on the time) cost 60 DKK and depart from Nyhavn.
Prices: Copenhagen is said to be an expensive city, but it really isn't that much more expensive than other large cities. Take a look at menus on the Danish realia page to see what how much a dinner would cost. Personally, I never spent more than 120 DKK on dinner, and usually only spent 35-50 DKK on lunch. Two restaurants I can recommend are both on Vesterbrogade; both are on the right side when coming from central station, a few blocks down and close to Hotel Løven and Hotel Savoy. Pizza / Pasta was only 52 DKK for a pizza margarita and 16 DKK for a small drink. Restaurant Ankara offers a nice buffet of Turkish food for 79 DKK, with drinks at about 21 DKK.
The official tourism site of Copenhagen is Wonderful Copenhagen.
My Map of Copenhagen:
View Copenhagen, Denmark in a larger map