For wine purchasing a good place is Rosforth & Rosforth. Even better is the location, under the knippelsbro bridge, very hidden.
Also nice for their professionally pleasant service and knowledge. And also for the interior.
This bridge leads from the center to Christianshavn and Amager. Christianshavn is a really sweet and old area, with a canal with lots of boats, nice restaurants and cafes. Right next to Christianshavn is the Freetown Christiania.
Copenhagen Meetpacking district is a good place to go out for a beer. Duing the summer the hole place is packed with young people. There is several bars, galleries and restaurants in the area.