Looking at a map of Copenhagen, you will find a lot of green spaces: we have a lot of recreational parks, where you can play ballgames, picnic, sun bathing, drink your coffee, limonade or beer (no restrictions). There are playgrouns for the kids in almost all the parks. The old defence area surrounding the old city has partly been made into parks, and it is also here you find the botanical gardens. http://botanik.snm.ku.dk/english/
The biggest park is Fælledparken at Østerbro, and the most interesting is Assistentens Kirkegaard at Nørrebro, which is a unique as it is both a graveyard and a park. Here you find the graves af notabilities like Hans Christian Andersen, Søren Kirkegaard and The guy behind the quant theory – Niels Bohr.
If you go a bit outside Copenhagen you can find the 6 forgotten giants, which is made by are artist, only using second hand materials. http://thomasdambo.com/works/forgotten-giants/
In the center you find the Kings gardens, where Rosenborg with the grown jewels are placed. It is good place to get away from the busy shopping streets. No biking in the park, and it is closed after dark.