Biking is the normal way of moving around in Copenhagen. Since you will be here for a limited period of time, we recommend you get a Swapfiets. These are rental bikes but you get your own bike for the time you are in Copenhagen. Swapfiets offers flexible schedules and student discounts. Just go to their website and make an appointment to pick up your bike.
Please always make sure that the lights on your bike are working. Not only is it for your own safety, but it is required by law, which means that the police can fine you 750 DKK if you do not have them.
And no worries, you can bike everywhere!