I'm Italian and I'm moving to Germany for work; I will also take my wife and 4 year old son.
How do I proceed to register residency? And once I register can my son go to school right away?
Details: Do I need a proper rent contract? Right now I don't have a rent contract since I've reserved an Airbnb; and I actually intend on staying in this same Airbnb at least until the end of the year (since my project is expected to end in December or January, if it extends or if I get another project or longer term employment then I'll probably look for a permanent place (buy or long'period rent) but right now I can't make a long term commitment).
As far as I know, EU citizens can stay in another EU country (e.g. Germany) up to three months without registering. If you know you want to stay longer, you have to register at the authorities of the town you live in, which are usually called Bürgeramt, Bürgerbüro or similar, and may be situated in the Stadthaus or Rathaus. You will find the address on your town's web site.
To register you need a Vermieterbescheinigung (if you have a rent contract). In my town it was also possible to let a third person confirm that you live in his or her house. Just ask for this form at the Bürgeramt.
The registration is the most important thing to do after arriving to Germany. Once you do it:
your children are elligible for German schools and adaptation courses
you can open bank account
you can register for social security and tax number (without it, you can't legally work in Germany)
However, if your children are not fluent in German, consider letting them stay in Italy until you know if you'll get a permanent position. Adapting to new language environment is quite stressful even for an adult. German schools are prepared to deal with children who are need to be taught German, but it might be difficult for a child being unable to freely communicate with other children.