You should try eating at Isted Grill, its a legendary place. Typical danish too in a sense. Also a place called Kebabish, its a little more expensive but its super good.
Of free things to do, i like to go down to go to a lot of the chillout spots, im sure you have been to Christiania by now? Or else, if the weather gets a little better soon, you could go to 'bongspottet' at Bryggen - its a little skatepark like place with a streety feel to it - and it has loads of people hanging out and a place to jump from and swim in the harbor (clean water though).
N?rrebroparken is nice, i usually try to go to Jarmers plads,it has that plaza feel to it.