Copenhagen is the happy and vibrant coastal capital of Denmark. The airport is only 15 minutes from city center by train or metro. Also, you can ferry in from many destinations. I suggest staying in Nyhavn, the colorful and picturesque 17th century canal waterfront. Hotel Bethel is my pick. It's great lodging for the price and puts you right next to countless eateries and photo opportunities.
A one-mile walk north of Nyhavn brings you to the Kastellet. Public transportation will take you about the same amount of time. If you arrive by boat, you can do this before heading to your lodging if you have light luggage since its near the cruise terminal. This star-shaped fortress from 1662 is one of the best preserved in Europe. Cross over the moat and stroll around the grounds to get an up close look at an old windmill and view some beautiful 1700s buildings. Outside the fortress to the northeast is the city’s most famous landmark, the Little Mermaid statue. The statue is actually pretty small and just offshore. There will undoubtedly be a number of people trying to get a perfect photo angle, although you may feel disappointed that this is all there is to it.
Walk south down the sidewalk from the Little Mermaid for a great photo near the Gefion Fountain and St. Alban’s Church. 10 minutes further south is Frederik’s Church, also known as the Marble Church. The green copper dome is a highlight of the city and can be accessed at 1:00PM. Directly down the road to the east is Amalienborg Palace. This home of the royal family is full of royal history. You can enter many rooms and also see the changing of the guard in the square. It is beautiful inside and worth a visit if you are a fan or royalty. However, I’d argue it’s not on the same level of grandeur as other royal residences you may be used to in Europe. For dinner, even though it's touristy, you can get very good seafood and pleasant outdoor seating at the restaurant row in Nyhavn.
You can make a day of attractions that are just west of Nyhavn. Start with a pretty 15-minute walk to Rosenborg Castle. It’s a stunning building that houses over 400 years of history. See Christian IV’s rooms, the coronation throne, a fine Venetian glass collection, and Denmark’s crown jewels. The Castle Gardens is the oldest and most popular park in Copenhagen. Although I always recommend eating local cuisine, Restaurant L’Anfora serves up good Italian offerings right across from the castle grounds.
The Botanical Gardens is nearby and features some interesting glass greenhouses. Or, you can head 15-minutes south right to Strøget, one of the largest and perhaps priciest shopping districts in the world. If that’s not your thing, just another 5-minutes south on the small island of Slotsholmen is Christianborg Palace. This is a must-see palace. Still a working government building, you can see such splendid ornate rooms as the Throne Room and Great Hall. The highlight for me was the Royal Stables. It was fun to visit the royal horses in their stalls and see the extensive carriage and equestrian tack museum.
Exit the island to the west to arrive at the National Museum and take a tour through the ages. I enjoyed the Viking Age artifacts and the well-preserved remains of early humans. For something to do at night, visit Tivoli Gardens just outside to the west. It is the second oldest amusement park in the world (the first is also in Denmark) and has attractions for all ages. I enjoyed the rollercoasters, the pretty gardens, and the way the park lit up at night. Expect a crowd. There are many fast food options of varying qualities and a brasserie, but I chose the Biergarten for Schnitzel and Tivoli Ale.
On another day, consider a side trip to Roskilde. It’s just a 30-minute train ride. One you disembark, a pretty one-mile walk north takes you to the Viking Ship Museum. You’ll see five Viking boats found preserved from the 11th century. Really cool! Outside, archaeological work is still ongoing. On the way back to the station, stop at Roskilde Cathedral, opened in 1175. You’ll see exquisite architecture and crypts that hold dozens of Danish royals. The cathedral even holds older burials, such as Bluetooth from 985.
Upon your return to Copenhagen, why not rent a bike? It’s the thing to do here and there’s less traffic to worry about if you head across the bridge from Nyhavn to Christiana. Freetown is Copenhagen’s hippie commune and they live by their own set of rules. Even if this isn’t for you, still come here to see the Church of Our Saviour. The highlight is climbing the church steeple, which has stairs on the outside! Not for anyone afraid of heights, it offers amazing views and is truly a unique experience. The climb may make you hungry, so it’s the perfect time for a 10-minute bike ride north to Reffen. This is the hip new street-food market. There are over 50 vendors with offerings from around the world. The food is excellent and it’s not uncommon to be among 5,000 people at any given time.