Exploring the region around Copenhagen can be super easy or require a little planning depending on your destination.
Copenhagen Central Station is a gateway to the region with a well-connected rail network to the rest of the region. The trains are comfortable to ride, but that means of transportation is not the only way to explore Copenhagen's beautiful countryside.
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There are good connections from Copenhagen by train. Especially from the Central Station and especially to the major towns and cities, but many trains stop at some of Copenhagen's other major stations like Nørreport, Østerport and Nordhavn too.
"Kystbanen" ("The Coast Line" is a regional railway going from Copenhagen to Elsinore. A beautiful trip and an easy way to explore this part of the region. Get off at Louisiana Museum of Contemporary Art or The Deer Park.
Travel by car
If connections to your destination are not good with public transport, consider going by own car or rent one at Drive On which is an app-based car rental service easy to use by foreignors. Like any other major city, you'll find the usual international car rental brands here as well.
Save money and get free public transport with a Copenhagen Card
With a Copenhagen Card you get free entrance to 87 attractions in an around Copenhagen as well as unlimited travel throughout the entire capital region. This means you don't have to worry about ticket or zones. Use their calculator to quickly see how much you're saving, if you plan to visit attractions in the city and beyond.