I get off the train in Snekkersten, a small coastal town 50km north of Copenhagen. The conductor has already warned me there will be nothing open: no food, no drink, no taxis. She is right. It is late at night. Really late. Close to midnight. I walk past the sleeping houses and along the beach. It is quiet down there. At my hotel, the Villa Brinkly, I find my room key inside an envelope pinned to the front door. A bicycle is waiting for me out there too, the key securely stored – in the bike’s lock.

Denmark is a safe country, consistently in the world’s top 10, with high levels of trust, rather than suspicion. It’s amazing how quickly you can relax into that – and wish the whole planet could be the same.

I am on the second week of my Scandinavian summer, now cycling around the coasts of Sealand (aka Zealand), where there is a new national park to explore.

In the morning, after a superb and much-needed breakfast in the hotel’s sunny morning room, I load up my bicycle and set out from the back of the hotel, directly into the forest. Today is intended to be a gentle warm-up, ending back at the same hotel. I am immediately launched on to a well-made trail through a shady beech forest, where deer stand in patches of sunlight and red squirrels stare down from above.

The trail is lovely: a rolling path that skirts fields of wheat, dives through patches of woodland, and passes a castle, eventually emerging on the coast. I take a detour to see the house of Karen Blixen in Rungsted. The desk where Out of Africa and Babette’s Feast were written is there, as are Blixen’s travelling trunk and letters. It’s a remarkable place, well worth a visit, with a good cafe too.

So too, reportedly, is the nearby Louisiana Museum of Modern Art, but in the heat I choose the meadow by the beach dotted with picnicking families, and go for a much-needed swim.

That evening I eat fish and chips sitting on the harbour wall in Espergærde, a short walk from my hotel. There is a fish kiosk there, and these offer an easy way to save money in what is the most affordable of the Scandinavian countries.

Next day I ride north up the coast, passing strings of thatched cottages, all of which manage to look gorgeous without being in the least bit twee. There is no attempt to ramp up the idyllic here, no gaudy hanging flower baskets or fake wishing wells. After a short time I reach Helsingør, a grand town hedged by fortifications and harbours. On a spit of land beyond the port is the imposing palace of Kronborg, a great square bastion with copper-clad towers, the castle where William Shakespeare set Hamlet. I cycle over and enter.

The bard never visited Elsinore (the anglicised name), but his friend and colleague Will Kempe certainly did, probably arriving in 1585, when King Frederick II was celebrating the complete transformation of his medieval fortress into a Renaissance palace. These days there are no ghosts on the battlements, just great views and a lively cast of actors.

The Maritime Museum next door is less satisfying: the building is a bold conversion of an old dry dock, but the thematic displays are thin on content. The best part, fortunately, is free: views of heritage boats assembled in front of a cafe. The town itself is worth exploring, with narrow streets packed with fascinating and colourful buildings.

Back on the trail, I turn west along the coast, which is lined with fine beaches backed by forests and meadows. I stop for another swim, then track down my digs, Badehotel, standing in clifftop splendour close to the town of Gilleleje. In typically understated fashion, it has no sign. Fortunately my cycling notes point out to me that it “has an anchor outside”.

Next day I head inland into the new national park, Royal North Sealand, which is really the former hunting grounds of another great castle, Frederiksborg. The path leads through deep forests of beech and oak, a glorious ride reputedly haunted by trolls. I do get a shock a few minutes in, but it’s only a goshawk, sweeping imperiously up the trail.

The park covers 323 sq km and is so new (inaugurated this spring) that as yet it has no signage, but I’m not sure the Danes would go in for them anyway. In places, I abandon the main trail to explore side routes, discovering groves of twisted ancient trees, and then emerge suddenly into the magnificent gardens of Frederiksborg Castle.

Here the ornate intricacies of the formal gardens are replicated in the interior designs, a reminder that Denmark was once both a great monarchy and a colonial power (at various times it held, among other places: Greenland, Iceland, the Shetlands, as well as parts of modern-day Ghana, Latvia, Estonia, India and the US Virgin Islands).

A royal castle is a curious spot at which to end my time in Denmark, a country that seems so defiantly against all privilege, hierarchy and ostentation. Some might find the Danish countryside a bit lacking in drama: there are no soaring mountains or dizzying gorges, nothing raucous or spectacular. What it offers is an opportunity to travel in a happy country, or at least one that is significantly more content than most.

The pleasure of that leaves a lasting impression, and raises all sorts of intriguing questions to while away my onward ferry journey to Sweden.

What is Denmark’s secret ingredient? Is it the lack of garish advertising and the quiet insistence on self-reliance? Or maybe it is nothing more than the gentle, good-natured buzz of their commonsensical society.

The memories of those things are, I reckon, better than anything from a touristy gift shop – which is handy, because now I think about it, I never saw a single souvenir on sale, not even a tiny gift bottle of liquefied hygge.

• The trip was provided by Inntravel, whose four-night cycling tour of North Sealand costs from £715 (including four nights’ accommodation, two dinners, four breakfasts, cycle hire, route maps, connections and back-up assistance). There are flights to Copenhagen with budget airlines from several UK airports

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