Difficult question, with no simple answer.
It is also a question that belongs in
Assumptions: All participants
parents, children, mother and sister
are Italian citizens.
All Italians living outside of Italy are required to register themselves with the AIRE system
if this is not the case, do so before leaving the ...
Your income tax and old-age pension will be paid in Germany and will be deducted from your pay automaticly.
When leaving Germany you should apply for a Versicherungsverlauf, which will assist you when applying for a Pension in the future
see details inside in the first link below, togeather with other tips one what must be done after arrival in Germany
You have said you are "moving" from Italy to Germany. That makes things much simpler. (Rules are more complex if you live in one country and work in another.)
Your salary will be taxed in Germany. You will not be liable for tax on your salary in Italy, provided you register with AIRE.
You should pay contributions for old-age pensions in Germany.