I highly encourage you to visit neighboring countries while in Europe: France, Switzerland, Austria, Germany, Greece, Croatia, and Spain. But, I also urge you to enjoy all the cities Italy has to offer.
A good, comprehensive list of must-see places in Italy: https://travel.usnews.com/rankings/best-places-to-visit-in-italy/
The photo book I created after my study abroad of all the places I went: https://www.canva.com/design/DAF3DQdevIE/Hf8A717DoDmaE71vLwF80Q/view?utm_content=DAF3DQdevIE&utm_campaign=designshare&utm_medium=link2&utm_source=uniquelinks&utlId=haa2b1cdfd8
Taking the train is often the most convenient and reliable way to travel from city to city. Italy has an extensive network of both fast-speed inter-city trains and regional services that connect the smaller cities and towns. Train tickets can be purchased in advance through an app called Trenitalia and are fairly cheap. Buses or private cars may be better for exploring the countryside. Italy has two main train companies. Trenitalia is the government-owned carrier that operates fast-speed inter-city rail and regional train services. Trenitalia was one of my most used apps while staying in Italy. Italo is a privately owned company that operates fast train services between major cities.