Here are some suggestions (to be extended):
Bikes, which can be rented at several places, and public transport are the way to get around in Copenhagen.
However, buying tickets for public transportation can be confusing since public transportation relies on a zone-system and there are different ways to acquire tickets. You can buy tickets at 7/11 stores, from ticket machines, and directly from the bus driver.
An excellent website in order to plan your journeys is rejseplanen.dk.
When searching for transportation in Copenhagen, you might stumble upon the City Pass which offers cheap prices for public transportation within Copenhagen for up to 72h.