Just 25 min by train from downtown Copenhagen lies an old and beautiful forest and country area inhabited by wild (but friendly) deer. It's a vast, mysterious space for getting lost in and perfect for spacing out on mushrooms if you bring friends. Stay away on Sundays as it fills up with screaming children and zombie parents. Grab an ice cream on your way back at Bakken, the old David Lynch-ish fairground located at the outskirts of the forest.
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.
Between colony gardens in Amager lies one of my favourite local spots: Baghaven. Good food and cosiness is in the forefront. Various events with and without music gather people on the square at Diamantgangen 75. A very local experience. Great for families, children, young and old.
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:)