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.

On the northern outskirts of the harbour, post-industrial Refshaleøen is a creative quarter with a growing food scene. Copenhagen’s buzziest neighbourhood. Thronging street food market Reffen (from March to September) offers budget-friendly options, while La Banchina is Copenhagen’s trendiest spot for sundowners (tip: bring a swimsuit for the sauna).

A gourmet food market that is a must-stop on the Copenhagen culinary trail. orvehallerne is a foodie’s paradise, its twin glass halls housing a cornucopia of seasonal fruit and veg, baked goods, fresh meat and seafood. There are also plenty of tempting food stalls where you can pick up a scrumptious dish or two to enjoy on the go or at one of the tables outside. Our favourites include gourmet porridge at Grød, confit duck sandwiches at Ma Poule and craft cocktails at Pavillon De Verre.

Possibly Copenhagen’s quirkiest attraction is this ski slope atop a tall waste-recycling plant. Designed by starchitect Bjarke Ingels, the artificial ski slope plus inner-city hiking area is the ultimate offbeat way to experience the city. If you don’t fancy adventure sports, you can walk up and stop for views of the Øresund strait as you go. Certified climbers can also have a go at the world’s highest climbing wall, measuring 85 metres from top to bottom.

From furniture and ceramics to industrial design and textiles, the very best in Danish design is showcased at this renovated museum. The rococo building used to house Denmark’s first modern public hospital before it was transformed into a museum in 1926. Pass through the museum halls and out on the other side to enter the open public garden that provides a mindful space in the centre of the city.