By train
The coastal railway Danish: Kystbanen runs along the Ãresund coast from Copenhagen and various destinations in southern Sweden though it will often be faster to take a train to Helsingborg and catch a ferry across the strait from destinations in Sweden. The trains are operated by DSBFirst (http://www.dsbfirst.dk), the journey takes 45 minutes from Copenhagen central station with trains every 10-20 minutes during the day and every hour during night time. Single tickets are very expensive 108 Kr in 2011, so consider using a 24 hour ticket, or if in a group an All Zones klippekort, with 10 trips for 455 Kr 2011 prices. If you are considering visiting the Louisiana art museum, it's both feasible and much more economical to lump them together in a single day trip.
The Lille Nord Little North railway (http://lokalbanen.dk/K%C3...) leaves once or twice an hour for Fredensborg and Hillerød, the sites of North Zealand's two other great palaces. Journey times are 25 and 33 minutes respectively, there is an S-Train back to Copenhagen from Hillerød, if you plan on seeing the palaces as a day trip. Finally Hornbækbanen runs along coast north west from the city to Gilleleje on the northern tip of Zealand, for access to the regions excellent beaches.
The train station in Elsinore, which has direct access to the ferry terminal, is actually somewhat of an attraction in itself. It hails back from 1891, and is build in the same style as the Rosenborg Palace in central Copenhagen. It was thoroughly renovated in the early nineties and subsequently listed. When arriving it's worth taking a bit of extra time to suck in the atmosphere of the richly ornamented and decorated foyer.
By car
The motorway E47/E55 from Copenhagen ends in Elsinore, it's a 50 kilometer 30 miles drive, taking around 40 minutes outside of rush hours. Route 6 takes you to Hillerød, in about 30 minutes.