Bologna is a great suggestion, since it is a college city. Trento is a student city as well. Ferrara is very open-minded, otherwise start going South and don't miss Perugia: good chocolate and awesome views!
When I was a student, my friends and I went to Italy for two weeks. During the first part of our trip we visited Milan, Verona, Venice, Metropolitan City of Bologna, Province of Florence, and Rome. We tried diverse Italian food and wine at different restaurants, enjoyed picturesque scenes around us.
Then we spend some days swimming and sunbathing at the spectacular Amalfi Coast and enjoy its kind sun and its unforgettable beaches.
A rental car really helped us to visit so many places in such a short period of time. The rental car cost is one of the lowest in Europe. We rented it in Milan at https://rental24h.com/italy and were really glad we did it. By the way, you can save some money while you have a car. We spent some nights traveling to another city. It is real fun since you are young.
Do check out Marche!
I would suggest Milan, because of incredibly busy both cultural and non/cultural environment...but it's a bit expensive, I must warn you. Maybe Siena, and also Bologna. It's not like in the other main cities (I'm thinking of Turin, Pisa, Lucca, Rome, or Napoli, for example) you won't find things to do :) Most of them are cities with universities, so you're sure to find a vibrant atmosphere!
Enjoy your stay here!
Italy has very beautiful cities that is hard to choose.
I like very much the Tuscany region, particularly Florence. There you can find beautiful beaches for summer holidays, and wide landscapes with vineyards. Otherwise you can try Pisa or Siena, that are also nice cities but smaller than Florence. In Tuscany there's so much tourism, so events are often organized, and the social life is very animated.
I would recommend Rome and Florence depending on what kind of nightlife you are looking for. Rome definitely offers more in terms of things to do and clubs to go to, but you won't feel as safe and the prices are more expensive than what you might find in Bologna or Florence.
I'd say for vibrancy, Rome. Job prospects, Bologna. Fun things to do and creative community, Florence.