Ny Carlsberg Glyptotek is one of Copenhagen’s most treasured museums, found right at the heart of the city. It’s been open to the public since 1897 and is renowned for its extensive collections of Mediterranean antiques and 19th-century art (among the biggest collections in the world). Exhibitions range from Ancient Egypt to the works of Manet and van Gogh, as well as sculptures from Greece, Denmark and France. Come for the art, and stay for the ambience.

Copenhagen’s most sizeable cultural attraction is an excellent way to acquaint yourself with some of Denmark’s best-known works of art, including the masterpieces of the Danish Golden Age and Vilhelm Hammershøi’s highly influential interiors of 19th-century Copenhagen. The 20th and 21st-century collection, housed in a striking modern extension, is excellent too. With its simple but scrumptious menu, SMK’s beautiful café provides the perfect pit stop.

Located in the city centre since 1870 and home to more than 13,000 plant species, this is the greenest place in town. It’s ideal for a leisurely stroll when you want to escape the crowds. Highlights include a rhododendron garden, rock gardens featuring plants native to Europe’s mountainous regions, and the Palm House. The historic victorian green house features a terrace from which there is a grand view of the entire garden. During summer, visitors can also explore the garden’s butterfly house.

Built as a summer residence for Christian IV, the spectacular Renaissance castle Rosenborg Slot is home to the Danish crown jewels, several lavish rooms and a priceless wine collection. The castle’s leafy surrounds are also delightful: the King’s Gardens (Kongens Have) is a much-loved green oasis where you’ll find a decent café, two pétanque pistes, a charming puppet theatre, and plenty of space for picnicking.

This museum is an impeccably designed, impeccably located museum with a world-beating modern art collection. Its permanent collection comprises more than 4,000 works produced since 1945, including pieces by Picasso and Giacometti, while its Sculpture Park houses work by Alexander Calder and Henry Moore. The building itself is considered a significant work of Danish modernist architecture.

The award-winning Superkilen celebrates the multicultural melting pot that is Norrebrø and exemplifies the power of inclusive design. Constructed in 2012, it stretches for close to a kilometre through one of Denmark’s most ethnically and economically diverse neighbourhoods and contains more than 60 objects that reflect the area’s diversity. Look out for swing benches from Iraq, a boxing ring from Thailand, a slide from Chernobyl and rubbish bins from Blackpool.

There’s fun for all the family at Copenhagen’s world-famous amusement park, which has stood on this site for more than 175 years and even inspired a certain Walt Disney. Adrenaline junkies will love the rides — including one of the world’s oldest wooden rollercoasters, constructed in 1914 — while others will find thrills in Tivoli’s seasonal decorations, floral displays, fountains, carousels, lanterns and old-fashioned carnival games.

Difficult to pronounce (try ‘smurr-brull’), Denmark’s classic open-faced sandwiches are a traditional lunchtime dish every visitor should try. Practically synonymous with Danish cuisine, smørrebrød comprises thin slices of rye bread topped with ingredients like pickled herring, battered plaice and chicken salad. Our favourite spots to try them are Kronborg, Schønnemann and Aamanns 1921.

One of the most enjoyable ways to see Copenhagen is by spending a couple of hours skippering a solar-powered boat, or joining a social sail with Hey Captain around the harbour. No experience is required to operate a GoBoat or a FriendShip, but you can opt for joining a social tour with Hey Captain if you prefer leaning back and making new friends instead of controlling the wheel. Remember to bring a picnic: GoBoat seats up to eight round a table.

This is Copenhagen’s most visited harbour bath, offering open-air swimming all year round. Denmark is full of harbour baths, and Islands Brygge has become one of the city’s most iconic spots for a swim. During the summer months, the harbour side gets packed with beachboys, regular swimmers and families alike. Swimmers can take in city skyline views and follow up with a picnic on the nearby grassy area. There are five pools, with the shallowest being 30cm deep.