After almost 9 years using my bike as the main transportation for everything, I am starting to feel confident. Not native, mind you, women my age around here have more than 50 years in the saddle, honing their thigh muscles, but no longer a tourist. And so I would like to offer some advice from somebody who still remembers how confusing this was.
Copenhagen has a wonderful network of bicycle tracks, and it's easy to rent bikes, manual and electric. I strongly recommend that you try this out at least for one day. However, Copenhagen traffic is complex, due to the mix of pedestrians, cars and bikes, and all the natives know the rules and get really upset when you don't follow them. So please remember:
1) You are operating a vehicle. A bike is heavy, and moves at reasonable speed, and that's why it is really important to follow the traffic rules for vehicles. Follow the driving direction. Don't drive on sidewalks. Don't drive on foot-paths. Don't drive on pedestrian crossings. Follow the direction of the other bikes, don't drive against the driving direction. Signal when you turn or stop. Vehicle, remember?
2) Your vehicle is small, slow and vulnerable. Don't try to be smart and outwit the cars. Busses and trucks turning right cause the most serious bike accidents in Denmark. Make sure they see you, make sure you follow the rules for bikes, don't confuse the drivers.
3) If you decide to behave like a pedestrian (this is the great thing with bikes, you can, easily), get off the bike. Either you are on it, and driving a vehicle, or you are off it, walking, and a pedestrian. And if that is what you choose to do, get out of the way, off the bike path, and preferably aside so you don't block the sidewalk.
That is basically it, but I have some extra hints:
The city center is a bad place to ride your bike. Getting past places like Nyhavn, Tivoli and Christiania is hard enough for the natives who know what to expect, with pedestrian tourists who forget that the extra sidewalks are actually bike lanes. If you try to do it, you don't know what to expect, and will end up crashing with some French lady too cool for rules.
Further, the city center has some of the most traffic heavy bike lanes in Copenhagen, and during rush hour people just want to get to the other side of the city, now. And if you, like the British dad I just almost forcibly met on the bike lane this morning, decide it is a brilliant idea to guide your little family against the driving direction of the lane, you are asking for trouble. The least will be a symphony of angry bells, the worst will be a multi-bike pile-up when the rush hits you head on.
Instead, anything that takes you out of the center is great. Take the bike to Refsehaleøen, it's a lovely trip across the canals and along some beautiful old roads, and keep going to hit the artificial beach at Amager Strand. Follow the canal south to the Royal Library and the Architecture center, and then turn back to find the Parliament at Christiansborg, or Tivoli. Go east, and circle Castellet, to reach the little Mermaid from the other side, rolling smoothly up like a native, instead of mingling with the less informed tourists. Or keep going north along the water, to the lovely, posh neighbourhoods to the north-east past Østerport, or to the beaches of Charlottenlund. Push out beyond Nørreport and towards Nørrebro, to visit H. C. Andersen's grave in one of the loveliest combined graveyards and public parks you may find (at least around here). This is when the bike is your friend, and will happily carry you out of trouble, well beyond the anger of annoyed Danish bikers and pedestrians, and the crush of all the other tourists who are not as smart and well informed as you.
Anyway - I don't expect tourists to be reading this. But perhaps you accidentally get a hit on google for this, and read all the way down here. If so: please, be safe, be patient, read up on rules, and check the map carefully. Knowing where you want to go will get you there much easier. Also: Pay attention to the traffic lights. I am probably that woman behind you, swearing because you were chatting and didn't catch the 7 seconds of green light for bikers in that specific lane. You just made me wait five more minutes, because yes, the traffic lights in Copenhagen are sometimes a bit impatient with error. If you do hear me muttering angrily under my breath as I am pushing past with my groceries, it's not personal. And I do know that not that many years ago, that was me, hesitating, and somebody else, grumpily muttering. Embrace the Copenhagen experience and learn the rules, and we will soon enough have you too being annoyed and using your bell to angrily scatter tourists.