Any help or advice you can give would be appreciated.

My name is Haleigh, and I am a U.S. Citizen living here in Milan. I have been here since September 2014 attending Universita Cattolica del Sacro Cuore for my masters degree in international business. I will complete my program in October 2015.

My permesso expires October 10 2015. And since I will no longer be a student at this time, I can not apply for a new student permesso/visa. However my family is coming in the end of October through November 15th to do some traveling in London, Paris, and Italy afterwards. I would like to join them along this journey, and I am wondering if there will be any problems in this since my Visa and Permesso will be expired?.  When I came here for exchange, I was told that after our visa/permesso expire, we have 90 days as a tourist to travel. So essentially I should be fine with the 90 days and my American Passport?.

After my permesso expires, do I have 90 days as a tourist? Or how soon after my permesso expires do I have to leave the country?
So will I be able to travel around the EU as a tourist, even though my permesso has expired?
Will there be any issues when I try to leave Italy?

I hope that I was able to describe my situation clearly! Any help or advice you can give would be appreciated.

Looking forward to a response.