It is of the utmost importance that you check which kind of IDs your destination Country requires, as well as medical documentation, otherwise you may risk to be rejected at local customs.
You must have a valid ID (no older than 5 years) on you to be shown to destination Customs authorities.
The following Countries allow free circulation within their borders: Italy - Austria - Belgium – France - Germany - Luxembourg - Nederland - Portugal - Spain - Denmark - Finland - Iceland - Norway - Sweden - Greece.
Other EU Countries require you have a valid ID card to enter (but not the Italian driving license).
Your passport (with proper Italian stamp) allow you to travel all over the world.