If you consider that Italy has only about sixty million inhabitants, a quarter of the population of the United States, it seems odd there would be that many cities worth visiting. Sadly, though, that is the case, and even to get a cursory glance at the most important cities, you’ll have to travel to lots of different places.
One thing helps you: every city except for Rome had one particular moment in time when it flourished and then faded, freezing the city mostly in a particular architectural stage. You can safely skip certain cities, say, if you are not interested in medieval architecture (but then, why would you not?).
Let’s categorize Italy’s cities by their offering.