Spent a week there, mostly just walked or biked around seeing the city, you can ride down to the little fishing village at Dragor, we also biked north up the coast. There are some cool little areas along it the coats , both north and south. Take lots of money, things are crazy expensive. There’s lots of royal places if you’re into that, we just walked around without going in. As you’ll see below my interest was seeing the bike infrastructure. Also you can take the train to Malmo in about 30 minutes.
FWIW, my little write up, but bike focused https://www.transportation-planning.com/blog/euro-trip-2017-part-2-the-grey-bike-lanes-of-copenhagen-denmark