With more commuters via bicycle in Greater Copenhagen than ALL of the United States, it's pretty obvious that this is a biking city. And unlike a country like Thailand, you'll have no worry about getting run over by a crowd of scooters or any other aspect of road safety. Even chaining your bike is generally unnecessary because of the city's low crime rate.
Bike rentals are available pretty much everywhere and afford you an opportunity to see the sites the same way that many Danes would in a safe and leisurely way. Home of the first large-scale bike sharing program on Earth, you can choose a bike from one of over a hundred kiosks and, after a refundable deposit, can just return it to a different one when you are finished.
There are numerous bike tours available, being that a big draw of exploring Copenhagen is its bike accessibility. Take a bike tour (Bike Mike is popular amongst tourists) or conquer your Copenhagen itinerary on your own; no matter how you do it, the flat topography and extensive cycling infrastructure make this the go-to mode of transportation.