I have been talking about s-tog and klippekort but I forgot to mention the basics. Where do you buy you klippekort? Well either from a shop inside the station or a vending machine directly on the platform. You can also find them in some newsagents.
Other essential info: if you ask kindly (to the vendor, not the vending machine of course ...), you may manage to get a card cover, some even have a line map on them! Classy and totally essential!