definitely rent a bike while you're there. this is by far the best way to get around in cph, both fun and efficient.
amager beach is really nice in the summertime. so are the parks, especially ørstedsparken + fælledparken. there are usually a lot of events going on in the parks during the summer.
i think it's too late for this, but the distortion street parties are fucking awesome! also might be too late, but roskilde festival is also amazing...
brunch / beer on the lakes is really nice. i'm partial to café 22.
vesterbro was my hood so i'm biased, but i think this is one of the coolest parts of copenhagen. especially along istedgade, you have tons of cool boutiques, cute cafés, etc.
in the kødbyen (meatpacking district) right next to vesterbro there are a ton of cool bars, clubs, and restaurants in an industrial setting. fiskebar is a great restaurant, and next door there's a nice bar with olafur eliasson lamps... lol. jolene used to be a really fun bar/club when i lived there, not sure about it anymore. but still worth checking out if you're in the area.
nørrebro in general is worth checking out, a lot of fun stuff going on there. bankeråt is an awesome place to have a dinner or a beer.
obviously christiania is a really fun place to wander around. make sure to really walk around and see the cool houses / trails around the area. the music venue "loppen" is fun and usually has great acts. there's also an awesome venue which looks like a movie theater, except the ground is all sand... forget the name though...
nyhavn is pretty touristy but also very cute, kinda necessary if you're visiting cph.
also fun to have a beer around the canals of christianshavn. really pretty area.
islands brygge is nice for a bbq/beer/swim.
of course i'm not going to mention all the obvious touristy stuff... strøget for shopping, rådhus, rundetårn, tivoli, little mermaid, etc... i'm sure you already know about all that and anyway i think they're the least interesting parts of copenhagen... though one super touristy thing i do suggest is to go on the boat tour, it's a great way to see copenhagen and get a bit of history in. whenever i had relatives visit i'd do the boat tour and have fun with it myself too ;)
the opera houses are really cool to check out. in general there's a lot of cool new architecture in copenhagen / ørstad, but that's a whole new post in itself...
louisiana is wonderful for contemporary art.
dragør is a fishing village only busride away from copenhagen, and it's ridiculously cute lol. i love going there.
have fun and enjoy all the hooooot beautiful scandinavian men ;) and be prepared for the dent in your wallet, shit's mad expensive there... if you're kinda on a budget the best bet is to just grab a sandwich or something from the supermarkets. but of course don't miss out on the flæskesteg, frikadeller, rugbrød, smørrebrød, and all the yummy fish stuff... mmmm. i'm missing denmark now...