The public transportation in Copenhagen is very reliable, punctual, and it takes you everywhere. In Copenhagen the trains, Metro and busses (including water busses) can be accessed with the same ticket.
The metro station is located by terminal 3, and the train runs every 5 minutes during the day and will get you to Kgs. Nytorv in about 15 minutes. From here you can switch to the M2 (circle line) which takes you all around Copenhagen.
The train station is located by terminal 3, and the train runs every 10 minutes during the day and will get you to Copenhagen Central Station in about 13 minutes. Tickets can be bought at the ticket machines in terminal 3 just above the railway or at the DSB ticket sales counter, also located in terminal 3.
Taxis can be found at terminal 1 and 3, and will get you to downtown Copenhagen in about 20 minutes depending on traffic. Taxis are usually waiting outside each terminal. The cost is approximately DKK 250-300.