Visit Copenhagen, the wonderful capital, one of the easiest cities in Europe to visit with a family, with walkers and cyclists outnumbering cars (it's home to the world's longest pedestrian street), pavement cafés, mime artists, small art galleries and hippy hang-outs, and some first-rate museums and shopping. Don't miss the world-famous Tivoli Gardens amusement park, Amalienborg Palace and statue of Hans Christian Andersen's Little Mermaid.
Head to Billund, home to the world's first Legoland, with its own hotel. Also easily accessible from the ferry port of Esbjerg is Ribe, Denmark's oldest city, which celebrated its 1300th birthday in 2010. As well as the well-preserved medieval centre with its half-timbered houses, cobbled streets and cathedral, it's worth visiting for the Ribe Viking Museum and Ribe Viking Centre.
Discover Arhus with its Viking origins, one of the liveliest and prettiest towns in Denmark and a good base – central, with very good transport links. Self-effacing, laidback and artistic, it has some excellent modern hotels and architecture.
Explore Funen, the third-largest of Denmark's islands, in the middle of the country. Known as the garden of Denmark for its fields of fruit trees and vegetables, it's a place of writers and artists (Hans Christian Andersen was from here), drawn by the soft light, lovely beaches and bucolic nature. It's best seen by bike.