Staying in Italy over 90 days and Visa (Rome)

Hello All,

I will be teaching in Rome for an American School based in USA for 3 months. My family and I would like to stay for 3 more months to travel in Italy and in other parts of Europe. I just learned of the 90 day limit and am very concerned. Does anyone have an idea of what kind of visa we could apply for so we can stay legally for the entire time.

Thank you