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.
Copenhagens bikelanes is all around the city. Its one of the best cities in the world to get around on bikes, there is even free bike servicestations around town:)
Ordrupgaard is a beautiful museum with lots of interesting exhibitions. Right next to the museum Finn jul's house is located, a famous danish designer and architect.
In the summertime you can take the Metro directly to the beach. And if you're over the crowded white trash tanning and blanket sex scene, enter the hidden away mens or womens area located in the big wooden beach bathhouse. Here you can lie naked and enjoy the sun on your private parts while reading a novel or surfing your iPad. Expect lots of gays and lesbos which can be a curse or a party depending on who you are as a person.