The 40 minutes train ride along the coast from Copenhagen is simple and scenic.
From Copenhagen Central Station the travel time is 45 minutes with trains departing every 20 minutes.
You can also get on the train from Nørreport Station and several others.
A an adult single ticket costs DKK 108, and DKK 54 for children.
At www.journeyplanner.dk you can easily plan your trip in and around Copenhagen and see current prices.
Going by car takes appromixately 30 to 60 minutes depending on traffic which can be relatively dense on the highway between Copenhagen and Elsinore in the morning and afternoon.
Check out these guides.