Of course you can travel a lot, but still the places you'll better know are those where you stop for a while. So, I can give you some suggestions about the cities where I've been living 'till now:

• Brescia (North Italy): that's the city where I grew up and lived for 21 years. It's a quite, nice, full-of-story city next to the Lake of Garda, famous for its natural beauty and for its products, like wine and olive oil. If you are in the nearby, don't miss the city downtown (with ancient roman relicts), the medioeval castle and the Santa Giulia museum, which every year holds a different world-wide famous collection. Tipical dishes from this area are essentially meat: try the spiedo, the tagliata and/or the salmi'
• Trieste (North-East Italy): although I've been living in this city for only 2 years, I'm deeply in love with it. It's a really beautiful, calm and imponent city next to the Adriatic see. In winter it can get quite cold when the Bora, a cold wind coming from the mountains, reaches up to 120 km/s, but the wheater is usually great thanks to the see. Despite the presence of the Adriatic see, the tipical dishes are not made by fish, but are more Austrian. Try the kren, a spices they use with meat, and don't miss the Osmiza in summer (small farms next to the city are open and you can eat and drink what they make for really cheap prices). If you want to visit the city, you must see the Miramare castle, Piazza Unita' d'Italia (especially at night) and the San Giusto Church. Last, try the coffee there: it's one of the best places in Italy for it. And they have literatly hundreds of different way to order it! If you are at the bar of the Economy department at the university, ask for a Capo in bi tanta special: it won't let you down.