When you are moving (in Denmark), you have 2 things to do, or I should say 2 places you have to declare it. To the Folkeregistret so you’ll get your new CPR card and to the Post. You can do both online in 5 minutes. And that’s it.
A few days ago I went on my online banking to change my info but they already had everything updated! That is just wonderful when you come from France, where everything takes weeks and never even work in the end.

If you are moving from Copenhagen to another place in Copenhagen, you go there.
If you are moving to another commune, you go to borger.dk, for example for me, Frederiksberg was here. But you can change the town on the upper right. Everything on this site is in Danish, so you have to chose “Anmeld flytning til folkeregistret” and use your nemID, and everything else should be pretty easy to understand, or Google Translate is enough. Because basically they only ask who is moving (if it’s only you or if you have a whole family following), your actual address and the new one. If there is already someone registered at this address and who stays there (for example you move in a shared-flat) you have one box to check.
You can choose to change your doctor too. You receive your new CPR card in a week.
You have 5 days after you moved to do this! But you can do it 2 or 3 weeks before already.

For the Post, you go there and follow “Gå til ePosthuset og meld flytning”. You also use your nemID. They ask if it is a definitive or temporary change, and you simply give your new address and the moving date (you have to do it 5 days before at least). It’s free. And it works well, I had a few letters redirected from the first day. And that too, is magical when one come from France 😉