Ciao and welcome to our community. For leaving the country, you need to follow the requirements for the country you're traveling to. Italy doesn't require an "exit Covid test." For example, I live in Rome and I recently flew to an EU country and since that country doesn't require a negative Covid test upon entry, I didn't need to bring one to the airport. On the other hand, US travelers returning home are required to have a negative Covid test to re-enter the country, so they have to present that when they check in for their departing flights from Italy.
In terms of entry into Italy (for anyone reading this who might be wondering), I spoke with an operator yesterday at the Italian health ministry's Covid/green pass hotline15 and she told me that people who don't have a booster shot but just the initial vaccination cycle have 270 days of validity on their vaccination certificate, according to Italian entry regulations. In terms of testing requirements for entry, it depends on where you're coming from. You can read more information about that here39.
You say six months have passed since the second dose. So long as it hasn't been more than 270 days since the second dose, according to this rule your vaccination certificate is still valid for entry into Italy.