A thing you have to try out is Absalon one of my favorite places in Copenhagen. For me Absalon is an oasis for freedom and relaxation and I will highly recommend to try their communal eating, it take place everyday at 6pm, but you have to be their before time to ensure that you get a seat.
Another thing you have to try is some of Absalon's events. Personally I love to play BINKO BANKO each thursday. You must keep yourself updated on their website because they update their calendar all the time, so you can see what is going on, what the food is ect.
So if you like to socialize, meet locals from Vesterbro, and want to be in a big living room - Absalon worth a visit
Check out their website: http://absaloncph.dk/