Based on many trips throughout my life.
I went to Italy many times as a kid and continued to explore the country with friends during my high school and university years.
Definitely one of the greatest cities in Europe. Enough amazing Roman architecture to keep you busy for many days (e.g. the Colloseum, the Catacombs, the beautiful mosaics in the spas near Rome). I had to study Latin for 7 years in high school and was immensely satisfied to finally get to use it (if only to read the inscriptions on old Roman buildings, grave stones etc.). Just as great as the remnants of ancient Rome is modern Rome: a bustling city with many nice squares and terrace cafes. When in Rome, also make sure you visit the Vatican which is again a world of its own.
One of my favourite cities in Europe, absolutely unique. I keep coming back and never get my fill of it. A maze of canals and small lanes, a car-free city where public transport is by boat, and an incredibly picturesque and romantic place. No wonder I picked Venice as the place to spend the Millenium (we had a great time, beautiful fireworks in an incredible setting, and happily singing Italians...however, apart from this exceptional year 2000, Venice is supposed to be pretty quiet on New Year's Eve).
Try to avoid the main tourist season in the summer (when it can also get very hot and sticky and the canals can quite smelly...). A great time to visit is in the fall (October or November), when it will often be foggy in the morning hours (adding to the romantic atmosphere), with a blue sky and nice warm temperatures for the rest of the day.
Apart from visiting the main sights like San Marco Cathedral (with amazing marble floors), the best way to experience Venice is by just walking around aimlessly. Just loose yourself in the maze of small streets and don't forget to explore the less touristy parts of the city. You should also take a boat to one of the nearby islands that are famous for their glass blowing art.
Venice is expensive, but you can easily get around those crazy hotel prices by staying at Archie's house (San Lenardo 1814B, within easy reach of the train station, Tel. +39 041 720 884). The hostel is basic but has lots of character. Best to show up early in the morning as they don't accept reservations.
Another famous Italian city. Beautiful architecture, but also very touristy, pretty much any time of the year. I particularly like the beautiful church San Miniato al Monte on top of a hill a little bit further away from the city center.
A very nice Italian town, famous for its open air operas in the ruins of the old Roman arena. Try to get tickets, it doesn't matter for which opera, as it's just one big party where people eat and drink during the "show" and singing vendors walk around during breaks (probably the perfect job for wanna-be opera singers). Great people watching.
A great part of Italy, with beautiful mountains and excellent hiking trails. The southern side of the Alps has a completely different feel to it than the northern side: a pleasant climate, orchards and vineyards in the valleys and great Italian food and wines. Particularly recommended in the fall when the weather north of the Alps might already start getting nasty while you can still enjoy a golden autumn in South Tirol.
A very popular place for camping and and outdoor sports (e.g. windsurfing, hiking, mountain biking). Beautiful scenery if you stay out of the main tourist areas.
My next trip to Italy is going to be to the Cinque Terre where I have wanted to go hiking for a long time...