Hey Angcox,
We went to Copenhagen about 3 years ago. There is a very convenient rail line from CPH airport to downtown.
We stayed in a Sofitel Plaza Hotel, near the central rail station and they gave us an airline rate of $80 a night. That is real cheap for that area! The hotel was near Tivoli and other attractions. It is a very expensive city, especially for food and drink. We bought most of our beer in a store instead of a bar. Public transportation is great and we did not need a taxi at all while there.
We also took the train from Copenhagen to Malmo, Sweden for a daytrip. It was not too expensive and took about 20-30 minutes. Malmo was a lot of fun too.
Let me know if you need any additional information.