I was fortunate enough to visit Copenhagen, Denmark for 5 days this past week and, spoiler, I fell in love with the city! I've mostly traveled in southern Europe (specifically in Italy), so this was completely new for me, but I highly recommend visiting the following neighborhoods and sites when in Copenhagen:
Nyhavn
Nyhavn is a neighborhood right on a canal with shops, street performers, and restaurants on either side. During the summer, Nyhavn is the place to be: you can grab a drink with a friend and sit under the sun, along the canal, sipping on a cold drink... until the ice melts and you get a refill, of course. There are locals and tourists alike, and it's a wonderful place to walk to see all the colorful, slated roof houses characteristic of Copenhagen. Plus, Hans Christian Anderson lived in Nyhavn for 20ish years, and you can see exactly where if interested.
Freetown Christiania
Christiania is a car-free neighborhood across the harbor from Nyhavn (about a 15 minute walk) and is home to around 1,000 residents. The reason "freetown" is in the name of the area is that this neighborhood is completely independent from Danish government.
Enter: "The Green Light District" aka Pusher Street aka there are small stalls with all different types of cannabis being sold. You can't take photos, so don't even try-- I did and was instructed to delete the photo while someone watched over my shoulder. So they're not messing around about the no photo policy. There are cafes, vegetarian restaurants, and music venues, plus beautifully groomed lawns and fields and tons of really great graffiti.
Christianshavn
This neighborhood contains Christiania, but it's so much more. It's composed of a handful of small islands separated by canals, turning a walk through Christianshavn into a romantic stroll. There are also large parks in Christianshavn with sunbathers, runners, bicyclists, and walkers, like myself (I never move above a brisk walk). Also, there's a casino here, if you're down to gamble.
Copenhagen City Center
The city center is a great place to visit at night to find popular restaurants and bars of every kind-- jazz bars, gay bars, dive bars, etc etc. A bit busier and more lively than the other neighborhoods mentioned, it's easily accessible by bus and you can of course walk everywhere. For a center of a city, it still has a lot of charm and character that you'll come to expect and love of Copenhagen.
Specific sites to visit
I am, as my dad describes, "artsy fartsy," (I have a degree in art history and have worked in galleries and co-ran an auction house) so I love to visit museums and art galleries everywhere I go. However, I know not all people enjoy this! So I will try to keep the overwhelming number of museum visits to a minimum in this list.
The Danish Museum of Art & Design (Designmuseum Danmark)
I heard someone describe this as "Ikea on steroids" but it was so much more than that! There were exhibits for interior design, fashion, and the historical progression for elements of design in Denmark. There were also things for kids to partake in, so it was suitable for everyone.
The Church of Our Saviour
If you dare, visit TCoOS and climb the spiral spire to reach the top. It's about 90 meters high and offers a great view of Copenhagen. My only advice is be prepared for sometimes steep flights, close quarters with other visitors, and if it's a hot day, heart palpitations.
The National Aquarium Denmark (Den Bla Planet)
I fell in love with this aquarium! A colorful coral reef, even more colorful fish, a sea otter presentation, and mermaids?! To say I was happy is an understatement. The aquarium is near the airport, so it's a good place to visit after checking out of your accommodation and before hoping on a flight home.
The Botanical Garden (Botanisk Have)
I felt that the Botanical Garden should have had more flora and fauna considering it was the middle of June, but it was free to visit so *shrug*. The reason this is in the "to visit" section and not below is because within the garden, there is the butterfly house which is lovely! There are so many different colors and sizes of butterflies that it's worth the extra cost to explore the butterfly house. See below for why:
What site to skip
I loved Copenhagen but let's be real: not everything can be #amazing.
The Little Mermaid Statue
I love The Little Mermaid, Hans Christian Anderson, and Disney just as much as the next 1990's kid, but this statue was just "meh." You have to climb down some rocks to reach the statue, and there is always a crowd of at least 20 people waiting for their photo op. It's a little removed from other sites and the city center, so for me it's a skip. Sorry Ariel.
So, there's my little guide for spending approximately 4-5 days in Copenhagen. If you're able, I recommend visiting this city as it's quickly become one of my favorites in Europe. Have a nice trip (or buon viaggio for my Italians, and god tur for my Danes)!