Copenhagen Airport sits on the Danish end of the Øresund Bridge, so Malmö and southern Sweden are a short, direct train ride away. The train leaves from the station in Terminal 3, a few minutes walk from the arrivals hall.
The Øresund train runs from the station in Terminal 3 across the Øresund Bridge to Malmö in about 20 minutes. It is the fastest and simplest way over to Sweden, straight from the airport without changing in central Copenhagen.
The same line continues past Malmö to Lund and on into southern Sweden, so you can stay on board for destinations beyond Malmö.
Travel to Sweden crosses a national border, so the 30 DKK Copenhagen 3-zone ticket only covers travel within the Copenhagen area and does not reach Malmö. You need a separate ticket for the crossing to Sweden.
If you would rather drive, a separate rank for Sweden-bound taxis sits outside Terminal 3, alongside the regular taxi rank.