Danish trains are enviably cheap, frequent and efficient. The Copenhagen Card, on sale at the airport, allows unlimited travel on the region's railways for 24, 48 or 72 hours, for pounds 14, pounds 25 or pounds 32 respectively - plus admission to many museums and other attractions.
Roskilde (36 minutes by train from the airport)
Currently celebrating its 1,000th birthday, Roskilde is a town emersed in history, particularly that of the Vikings. During the Fifties, Roskilde fjord was discovered to contain five Norwegian ships. This amazing find was excavated in 1962 and the ships are now the centre-piece of the Vikingskibshallen museum. No new wood has been used. It is awe-inspiring to be confronted by the kinds of warship once used to terrorise the English. Also worth seeing here is Roskilde Domkirke, the medieval cathedral and burial place of 38 Danish monarchs.
Elsinore (1 hour 10minutes)
There's one irresistible reason for visiting this busy port, and that is Hamlet. Although there's no evidence that Shakespeare ever in fact visited Elsinore (or Helsingor) Castle (Kronborg Slot), he used it as the setting for the famous play.
Hamlet, the Prince of Denmark, was, of course, a fictional character - perhaps the most famous Dane that never was. Elsinore Castle, which dates back to 1420, was restored as a museum in 1922; now it's one of Scandinavia's most visited sites and has provided the backdrop for several Hamlet film adaptations.
If you climb the castle's ramparts you can see across the strait to Sweden. Climb back down, and you're likely to bump into gaggles of drunken Swedes attracted (incredibly, given its already steep price) by "cheap" beer. If you've got more time, the Tecniske Museum displays the world's first tape-recorder and Europe's first successful plane (it was airborne for 11 seconds) - all for only pounds 2.
Round off with an ice-cream from the parlour called Brostrde-Is on Brostrde, one of the narrow, cobbled streets that lead down to the harbour, then watch the yachts cruise by.
Arken (reached by S-train to Ishoj - 45 minutes - and bus)
This is something fresh in the state of Denmark, built for Copenhagen's tenure as European capital of culture in 1996. Set amid a landscape of sand-dunes, this ark-shaped temple to modern art and architecture has been home to Danish, Nordic and international art since 1945. Innovative new media such as video and installations are well represented. The main gallery is filled with light - even on dull Danish afternoons - thanks to its sloping glass roof, making the Arken an uplifting experience.
Louisiana (1 hour)
This modern art museum is one of Europe's best and certainly one of the best located - aloft on a spectacular clifftop from whence a pathway leads down to a small, pebbly beach.
The outdoor sculpture park boasts the brooding Baltic sea as its superb backdrop. Although the galleries themselves may initially disappoint, persist and you'll find you are highly rewarded: Pablo Picasso, Andy Warhol and Francis Bacon are here. Don't miss the Mir exhibition, which runs until the end of this year.
Flying to Copenhagen
There are plenty of deals from London's airports, eg the pounds 50 return on Go (0845 6054321) for midweek travel from Stansted. From other points, fares are higher; British Midland (0345 554554) flies from Edinburgh and Glasgow; BA (0345 222111) from Birmingham; SAS (0845 6072772) from Manchester.