It's a great city to walk around.
Nyhaven is really nice .... obviously popular with tourists, but also locals. If you can get a good table, and the weather is decent, it's a great place to sit with a drink, order up some seafood snacks and people watch for an afternoon. From Nyhaven there's a nice circular walk ... head south east to the main waterway, hang a left and just keep following on the waterside path out to the Little Mermaid, where you will be distinctly underwhelmed. But, right next to it, is the old forts's earthworks... you can walk back towards town through there ... sort of a pleasaant park now. Head for the Geffion sculpture/fountain and the Anglican cathedral. From there you can take any of the parallel roads back to Kongens Nytorv.
Amergatorv is the main pedestrianised shopping street ... there's a good cafe to sit outside and people watch by the Stork fountain. That main drag runs between the town hall square and Kongens Nytorv.
There's Tivoli gardens (wasn't a huge fan), but around there you have a food hall and some good foodie outlets.
I enjoyed the walk up into Fredriksberg ... from there I walked across to Ravnsborgade, which is a bit boho antiques, book stores, charity/thrift stores area. The back down through the parks with the National Gallery, Botanical gardens and Rosenborg castle.
It's feels to be a very safe city and the nightlife is lively on the streets. Check youtube...there's a channel that did loads of walking around Copenhagen, day and night ... showing best routes and good night spots etc.
It's a great city to just look at the maps and head off on foot... going where the fancy takes you.
Watch out for bikes.
