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.

Ancient History
That's maybe the most important category for the sight-seer. In this case, most of the action is South of the Appennines. Tuscany offers the Etruscan cities, with wonderful tombs that are rich in decoration. Lazio has Rome, of course, a gem if you've ever seen one. Here the antiquities are for everyone to touch: I'll never forget how we used to make out in the baths in front of the Colosseum, with the ribs of the building creating cozy spaces into which only one car would fit, tight enough to ensure nobody would be able to open the doors. Further South, there are the Greek cities of Calabria and Sicily. Agrigento and its Valley of Temples is stunning, particularly in the late winter when the almond trees bloom. Sardinia has its own pre-historic culture that built houses and forts of stone. The complex at Barumini will haunt you, especially when you compare it to the dwellings of the Pueblo Indians.
The Middle Ages
Sadly, Italy is full with the remnants of medieval times. You'll run into a church built in the XII century anywhere, and it is quite saddening to see all this history wasted in one place... Let's see the castles of Trentino and Aosta, perched up on the Alpine foothills, cannons feeding greedy lords. Don't miss the stunning beauty Ravenna, with its wonderful churches encrusted with mosaics that I am sure you have already seen in some book or movie. Take your time to see medieval Florence, and Pisa. Yes, you may not know, but the famous Leaning Tower was built in the XIII century, almost eight-hundred years ago. Go to Perugia and Assisi, don't miss the castle of Frederic II in Apulia, stop in Amalfi, check the cathedral in Palermo. Too much to see, I told you!
Renaissance
Odd we would use a French word to name a phenomenon that is essentially of Italian origin. Rinascimento, re-birth, is the elation that came to the people of Italy after the Dark Ages ended. This is when the arts and the crafts flourished, when the merchants of Venice and Florence created the banking system and made themselves the richest men and women in Europe and thus in the world. The word 'bank' itself comes from Italian banco, bench or bar, where the Medici would do business. You hear, go to Florence for the Renaissance. Follow Michelangelo (late in the age), Raffael and the others to Rome, where you'll find more sculpture and painting than Renaissance architecture. And follow up with a visit to Venice and the Venetian hinterland, where the bloom was later, but the peculiar conservatism of the powerful aristocracy forced conformation with earlier styles.
Baroque
A time of deep depression for most of Italy, the Baroque era left its marks mainly in Rome and Naples. Rome is covered with churches with Baroque facades, Il Gesu' and of course the great Basilica of St. Peter being the finest examples. If you love the riches of Baroque art, stop at the many places that have Caravaggio paintings on display.
Modern Ages
Here it starts getting ugly. Italy got into a deep slump after the XVII century, and not until the nation was united in a struggle against the French and the Austrians would the situation improve. The French and the Austrians, ever since Napoleon, were caught in their own struggles, so that it was just for the Kings of Sardinia (who actually resided in Turin, which is not at all in Sardinia) to conquer the recalcitrant country, oust the ever since sulking Pope and build ugly many-storied buildings all over the country. At least industry started picking up and the Nation got its much needed infrastructure.
Contemporary Art
Since the turn of the XIX century, Italy dramatically changed again. Milan started becoming a hot-spot of the arts, where everything was going on, especially music. It is at the Scala that Verdi would show off his latest tunes, and it is here that fashion was made an industry. In the XX century, Milan kept its primate. Rome caught up partially with the advent of Fascism, but this proved to be a very short-lived intermezzo. For some outlandish Fascist architecture, visit EUR, the town built by Mussolini for the Expo.
Next: And if you don't care for art?