The most famous attraction in Helsignør (in English, Elsinore) is Kronburg, the Renaissance castle that overlooks the Øresund which separates Denmark and Sweden. It was here that Shakespeare's Hamlet wandered the windswept ramparts at night. Even if you aren't interested in its connections to Hamlet, the majesty of the late 16th-century fortress justifies a day trip from Copenhagen.
The small town is also home to a more recent attraction, the M/S Maritime Museum. The museum was designed by Denmark's most famous architecture firm, BIG (the Bjarke Ingels Group) and opened in 2013. The museum sits in one of the town's old dry docks—you'll walk over it as you make your way to the castle. The exhibits themselves are largely kid-friendly introductions to the sailor's life, complete with a station where you can apply your own nautical themed tattoo (don't worry parents, they are temporary). Some displays are surprisingly graphic and frank, like those covering the incidents of venereal diseases among sailors. The museum concludes with a celebration of Danish maritime might where you'll learn that one-tenth of the world's shipping today is on Danish vessels and you can track the country's cargo vessels on an interactive world map.