Welcome Lee!
As you have probably noticed, the Scandinavia section of Scubaboard is pretty quiet. You will have more luck on the Danish
www.dykkerforum.dk. Even though the conversation there is mostly in Danish, I am sure that you will get replies in English if you ask in English.
As for clubs, there are basically 2 types that you can choose:
1) a dive center affiliated club
2) a non-profit club
With option 2) I don't have personal experience. I have heard good things about "Vedbæk Sportsdykkerklub" which is not too far from you. As I understand it, the training that you can get there is CMAS, not PADI. Nothing wrong with that, but you specifically asked for PADI. Other people on dykkerforum.dk can give you more advice about clubs.
For option 1), I know that several of the dive centers around Copenhagen also run an affiliated dive club. The benefit of that is that there will be no mandatory volunteer work to be done by the members and the downside is that it will be somewhat more expensive, as the purpose of the club is to somehow drive some profit to the dive center.
I know 2 dive centers that I like and use: Scubagear on Amager (
http://www.scubagear.dk). This is a very personal and service minded shop. The teach SSI courses. For PADI, I use Master Dive in Helsingør. I have just completed my PADI IDC with them. I live in Lyngby.
You might also want to check out the facebook group "Dykkere i Københavnsområdet":
https://www.facebook.com/groups/dykning/
I hope you enjoy Copenhagen!
Mikkel