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It's a far cry from Las Vegas, but it used to be the only casino in the country from its beginning in 1903 until 1990, and it is still one of only 3 in the country. It's in a historic beach resort, with great views. You have to be 18 years old, and you must present a valid passport or driving license at the entrance. Dress code exists, though it's fairly relaxed, just avoid showing up in your newest Adidas track pants.
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Marienlyst Palace
Built as a royal pavilion in the 16th century for the king to enjoy the amazing vista over the sound, the residence was extended in the 18th century in neoclassical style and served a great variety of tenants before being taken over by the municipality in the 1930s. The grand beautiful interiors are supplemented by a nice collection of art with local motifs and silverware. Following the public takeover the garden was redesigned to closer match its neoclassical origins, and ensure public access to the great views over Øresund.
Øresund Aquarium
Small salt-water aquarium exhibiting local sea life. Unless you have a general interest in marine biology you can probably give it a miss, but they are great with kids - the university of Copenhagen staff are great intermediaries, and will probably be able to handle foreign kids, since many of the activities, like the feedings, are self-explanatory hands-on activities.
The Culture Yard
The newest addition to the city's cultural life is an interesting modern architectural piece fitted into the old shipyard overlooking the castle. The house features two stages for the local theater and frequent concerts, an exhibition space for changing displays of contemporary art and photography, the city's main library with a small English section, internet access and various events held regularly.
Shipyard Museum
In the old shipyard administration offices a group of passionate previous employees together with the city council have set up an exhibition on shipbuilding in the city including collections on the craft in general.
Carmelite Priory
Founded in 1430, and the finest example of a complete monastic complex surviving in Denmark, and one of the best in all of Scandinavia. The Carmelites were invited by the king to enhance education and knowledge in the city, and more importantly care for sick sailors passing through the city. When the order was expelled from Denmark following the reformation, it was turned into a hospital, but thorough restoration works throughout the 20th century have preserved the premises for posterity.
The Danish Maritime Museum, M/S Maritime Museum
It's all about Danish maritime history since the renaissance, told by photographs, models and exhibits on everything from the Danish colonies in India, Africa and West Indies over naval battles and the east Asia trade to recent history.