Another look at the map, and you’ll spot something relatively new: a straight line across the channel. For the first time since the last Ice Age there is now a dry foot – well, tyre – connection between the two Scandinavian countries, in the extraordinary shape of the Øresund Bridge. But you probably won’t want to leave Copenhagen in the first place. After all, its medieval centre is perfectly scaled for strolling, then modern planners got in on the act by pedestrianising Europe’s longest shopping street, Strøget. Couple that with a child-friendly culture and an unusually handsome people, who can mostly speak near-perfect English, and you have possibly the perfect weekend break.

The Danes number only some five million, located visibly on the margins of Europe, and they share with that other outlying nation, Britain, a particular scepticism of the bureaucratic tendencies of Brussels. Having already rejected the euro, the Danes would really rather not be assimilated, thank you very much. And why would they need to be? Denmark is one of the richest countries in the world, though it’s a place that wears its wealth lightly, and that seems to avoid displays of conspicuous consumption.

The city’s architecture soars from medieval to rococo, and style varies from eclecticism to the cool designs of Arne Jacobsen. The impact Danes have had on the world of contemporary interiors is second only to their expertise in fairytales.

You would need several weekends to cover all of Copenhagen’s sights, but to make the most of a single visit, invest in a Copenhagen Card (available from the tourist office; see below), which offers free transport and access to museums and attractions. Some museums are free anyway and most don’t charge entry fees on Wednesdays.

Tivoli (Vesterbrogade 3, 33 15 10 01, www.tivoli.dk) holds a near-mystical place in the affection of the Danes. It is as if each and every one of them is umbilically linked to the place and they never quite shrug off the childhood urge to make the pilgrimage when it opens in April. So what’s so special about this relatively small pleasure garden jammed into the centre of town? To find out, you’ll need to visit as night falls. In the daylight Tivoli is charming, with its lake, rides and magnificent flowerbeds. But in the evening there’s a transformation from amusement park to dreamland, into something, dare we say, magical. However, in this fallen world, magic doesn’t come cheap. Entrance is expensive, and you still have to pay for the rides once you get inside. A look at the restaurant menus might also persuade you that a picnic is an excellent idea.

Adjoining Tivoli, just off the main square of Rådhuspladsen, is Copenhagen’s crowning artistic treasure, Ny Carlsberg Glyptotek (Dantes Plads 7, 33 41 81 41, www.glyptoteket.dk). In case you’re wondering, yes, beer money did fund the museum. But even a short visit will show the cash has been well spent. There’s an extraordinary collection of ancient sculpture, with the largest array of Etruscan art outside Italy, as well as 35 works by Paul Gauguin to complement a selection of Impressionists, and 30 of Rodin’s sculptures. Don’t miss the glorious steamy glass dome of the Winter Garden either, with its huge subtropical plants and beautiful fountain. Some parts of the museum are closing for renovation, so call if you have specific areas of interest.

Nearby is another world-class museum, the Nationalmuseet (Frederiksholms Kanal 12, 33 13 44 11, www.natmus.dk), with its excellent Prehistoric, Medieval and Renaissance departments. Look out for the gold statue of a horse pulling the sun chariot, ploughed up by a Trundholm farmer in 1902 and believed to date from around 1200 BC. Further evidence of Danish concern for children is provided by the basement Children’s Museum.

However, where Copenhagen does fall down is when it comes to commemorating its most famous former resident, Hans Christian Andersen. There’s the pitiful display in the Ripley’s Believe It or Not (Rådhuspladsen 57, 33 32 31 31, www.ripleys.dk) and that’s it. Fans of the great writer would better spend their time rereading his stories, or taking a trip to Odense, his birthplace (see below).

Those with plastic to wield should start strolling down Strøget, the main pedrestrian shopping strip. To the north of Strøget is the old university area, or Latin Quarter, made up of narrow alleyways and lively squares, which fill up with tables and parasols in the summer.

North-east of Kongens Nytorv is the pretty canal of Nyhavn, lined with bars and restaurants, and the departure point for sightseeing cruises. Prices are set with tourists in mind, and locals bring their own beer and sit on the cobblestones, their feet dangling off the quay. It was here, 150 years ago, that Hans Christian Andersen wrote many of his fairy stories. The inexplicably famous Little Mermaid statue sits meekly further north on the waterfront, along the Langelinie docks. It seems, though, that the locals are sick of their symbol. She has been painted red twice, had her head hacked off three times, an arm lopped off once and, in 2003, she was actually blown off her rock by a bomb. So, if nothing else, you should visit her to check if she’s still in one piece.

Back in town, just over the water, is the island of Slotsholmen, the ancient heart of the city. It is laden with important buildings: Christiansborg Slot (Prins Jørgens Gård, 33 92 64 92), a fairly grim royal palace that now houses the Parliament (Folketinget), as well as the Supreme Court, the Royal Stables and the Armoury Museum, and the Black Diamond (Søren Kierkegaards Plads 1, 33 47 47 47), a fine piece of modern architecture and the home of the national library. The Royal Library Garden houses the brand new Jewish Museum (Proviantpassagen 6, 33 11 22 18, www.jewmus.dk), which was designed by Daniel Libeskind.

Further north is the stately quarter of Frederiksstaden, delineated by wide, straight streets and dotted with churches, palaces and museums. Here you will find Amalienborg Slot, the Danish royal family’s abode, comprising four rococo royal palaces. A raised flag shows if the first family is at home. If so, the changing of the guard takes place at 11.30am. Nearby is Copenhagen’s main defence, Kastellet (33 47 95 00), a chunky, star-shaped fortress located just by the Langelinie docks.

Ten minutes inland, Rosenborg contains a classic piece of royal architecture, the Rosenborg Palace (Øster Voldgade 4A, 33 15 32 86), which lounges regally among lovely gardens. Nearby, the National Gallery (Statens Museum For Kunst, Sølvgade 48-50, 33 74 84 94, www.smk.dk) has a vast collection depicting 700 years of Danish art from 14th-century pictorial work to the most contemporary installations, plus pieces by Van Gogh, Matisse, Picasso and many Dutch masters.

Across the water is Christianshavn, an expensively bohemian run of canals and cutesy Danish housing. The centrepiece of its skyline is the weird twisting golden staircase of Vor Frelsers church (Sankt Annægade 29, 32 57 27 98, www.vonfrelserskirke.dk). The top provides visitors with a grand view of Christiania (Bådsmandsstræde), the ‘free city’, whih has been squatted since the 1970s and whose future is a matter of keen debate. Nearby, just west of the centre, the fast-gentrifying Vesterbro was once the red light district, as illustrated in its City Museum (Københavns Bymuseet, Vesterbrogade 59, 33 21 07 72, www.bymuseum.dk). The Carlsberg Brewery (Gamle Carlsberg Vej 11, 33 27 13 14, www.carlsberg.dk) has free tours ending with a tasting session.

• Tourist information: Vesterbrogade 4A (70 22 24 42, www.visitcopenhagen.dk).