Palermo is the capital of Sicily, the large island in the South. The people call themselves Sicilians rather than Italians. Palermo is known for the large cathedral, Sicily is known for wine, olives and ... the Mafia (~movie The Godfather).
Alberobello is a small town near Bari in southern Italy. It is famous for its unique trullo buildings.
The Amalfi Coast is very picturesque and lies on a steep southern shore of the Sorrentino peninsula.
Bolzano is the capital city of the province of South Tyrol and by far the largest city in this Italian region. This major city is close to the very picturesque Dolomites mountains.
Cinque Terre ("5 Cities") is a rugged coastal area in the North-West of Italy. It is very scenic and one of the best places for a few-days hike.
Florence is a city with a long history. It was the home of Michelangelo and Leonarda da Vinci. The two major tourist attractions are the Dome and the bridge, but we can also recommend the delicious ice cream of which there are many flavours here!
Lago di Garda is the largest lake in Italy just south of the Alps with many resorts, camp sites and tourist locations.
Milan is the "capital" of Northern Italy and a very influential city when it comes to economy, fashion or music. A highlight could be a visit to the Scala, if you can get tickets,