I lived in Copenhagen for several years, but that was over a decade ago so forgive my scattered thoughts. I was back to visit last summer, and the city was as pleasant as ever. Overall it is a safe and friendly city, even if it looks a little on the scruffy side. People are welcoming and speak excellent English, including the latest US slang (expect to overhear a good dose of cursing). Public transit is excellent - there is a self-driving train from the airport to center city that costs around $5 IIRC, taxi drivers are usually nice, and they have city bikes for rent. You'll see people biking and bikes parked everywhere, and watch out for them as a pedestrian!
I hope you aren't on a budget -- it is a pricy city. Keep some spare change handy for restroom visits. On a first visit I recommend the hop on/hop off bus/boat tour that gives a good overview of the city.
Bring comfortable walking shoes for cobblestones and leave a few hours to enjoy Strøget ('stroh-yay'), the pedestrian mall that meanders down different connected streets. There are general department stores, and Illums Bolighus features several floors of the best of Danish design. There is a nice view over the top of the city if you walk up to the top of the round tower Rundetaarn. Duck down the side streets for bookstores, antique shops, or cool office supplies.
The patisserie La Glace is a charming old fashioned tea room.
If the day is warm enjoy a soft-ice-cream cone (much creamier than the US versions) from a street stand (chocolate sprinkles - druse - are also recommended). Many shops offer sandwiches to go so you can enjoy your lunch on a bench outside to people watch. You'll find botanic gardens and rose gardens for strolling. Visit Nyhavn of course. The Little Mermaid statue is underwhelming (and on at least her second replacement head :-/) but is on the list of things to do and near my favorite fountain the Geifion fountain.
Tivoli Gardens is an old-fashioned amusement park (no neon or plastic!) in the heart of downtown that is a nice place to spend an evening. For real cultural immersion you can check out Freetown Christiania, a hippie commune near the downtown. They've cleaned up their act lately, but I remember stalls openly selling several varieties of pot.
