The Dyrehaven Deer Forest and the Bakken Amusement Park
Trains run regularly to Klampenborg North of Copenhagen (line C). The trip only takes approx. 20 mins. Just outside Klampenborg Station to the left, you will find the gates to the vast Dyrehaven forest - an ideal spot for a picnic. There are also horse-drawn carriages available at the station if you feel like venturing deeper into the forest. The forest teems with deer, and you should be careful not to get too close to the animals during the mating-season.
The Bakken Amusement Park is also situated in the Dyrehaven forest.
If you turn right at the station, you will see the Bellevue beach. There is a good chance you won't have the beach to yourself on hot summer days, the place is usually crowded and great for people-watching. The quality of the water is good.
The CPH Card covers your train fare. The CPH Card can be purchased at the Visitors Centre, at major hotels, and stations.