There is a really good restaurant called Italia. They serve pasta and pizza, but also steaks and seafood. I go there all the time. The service is great, and it's not as expensive as some other restaurants. It's my favorite place.
Den Gamle By - It means 'Old Town' in Danish, and buildings there date back to 1597. People like to go visit that part of town.
Go to a game at a NRGi Park. It's a good stadium with good fans and a great environment. And you can watch me play!
Hit the cafes on Aaboulavarden by the canal. There's usually a good group of people out at night. Everyone speaks english, so it's easy to meet people. And they serve really good coffee - and other things, too.
There a loads of biking trails, many of which lead to the beach about 20 minutes away. Best to go in summertime!