The Year of Living Danishly by Helen Russell (2015) — A British journalist moves to rural Jutland and spends a year investigating what makes Danes the world's happiest people. Warm, funny, and genuinely informative about Danish society. A good read before you go.
The Almost Nearly Perfect People by Michael Booth (2014) — A British writer examines the Nordic countries with a fond but unsentimental eye. The Denmark sections are sharp on hygge, Janteloven, and the gap between the country's self-image and reality.
Smilla's Sense of Snow by Peter Høeg (1992) — A thriller set in Copenhagen, involving a Greenlander woman investigating a child's death. Atmospheric, dark, and a reminder that Denmark's relationship with Greenland is complicated.
The Invention of Nature by Andrea Wulf — Not Denmark-specific, but Alexander von Humboldt had significant Danish connections and the book illuminates the era when Copenhagen was a centre of natural science.
Number the Stars by Lois Lowry (1989) — A young adult novel about the Danish rescue of Jewish citizens during the Nazi occupation. Historically accurate and moving; the Danish rescue operation — in which the population helped ferry roughly 7,000 Jews to neutral Sweden in fishing boats — is one of the more remarkable stories of the war.
Hamlet by William Shakespeare — It's set here. Kronborg Castle in Helsingør will make more sense if you've read it.
For travel guides, the Lonely Planet Denmark and Rough Guide to Denmark are both solid. Rick Steves covers Denmark well in his Scandinavia guide, with emphasis on practical self-guided walking tours of Copenhagen and Helsingør.