Well, I actually just moved back from CPH. I lived there for over a year and had a great time. There are definitely some positves and negatives to CPH. The positives are everyone speaks English, likes Americans, and are usually willing to help you anytime. The negatives are the tax system, lack of sunlight from mid November to early February not to mention it's cold as hell. If you visit CPH definitely do not go before May. I highly advise going for Min Sommer holiday which is in either June or July. Also, definitely go accross the bridge to Malmo. It's Sweden's third largest city and is completely different from DK which is interesting because it's only a 35 minute train ride. Anyways, there is a lot to do in DK but there aren't a lot of historic monuments that you would know about other than the little mermaid which by the way is super small and not very impressive. Check it out and just try to understand what people are saying. You're surely in for a good laugh and fun times!