Hi Zoya. Italy is full of great things to see, and even after 2 weeks there will still be lots to come back to!
In the North, I agree with Julia, the Cinque Terre is absolutely lovely, and well worth spending a few days at. The 5 towns themselves are lovely, but the best bits are walking along the coast between them all. So pretty!
I would also suggest if you have time visiting one or some of the lakes - e.g. Lake Garda. From here it is easy to visit Verona. Don't bother with Romeo and Juliet's balcony (way too touristy), but do wander the streets and walk out of town up the hill to look back onto the roofs below. Also if you are even vaguely into opera, try to get into the amphitheatre in Piazza Bra for one of the concerts during the opera festival - excellent music/singing and you can take in your own picnic!
And then of course the north would not be complete without a trip to Venice. A truly wonderful city, with wonderful architecture, unique canals, etc, etc! On your way south, drop into Perugia. This is a pretty town on top of a hill. They have a good jazz festival on in the summer and lots of nice streets to wander through. And then there is Siena (not sure about that spelling). This is where the Palio horse race is on (which you will probably be too late for, but they still have lots of processions through the streets afterwards). Oh, and I almost forgot Bologna! Bologna is a great place - very studenty feel - lots of good bars, great cocktails! Very difficult to get food there though at 3pm.
Then you are down at Rome. Obviously a wonderful city with plenty to see. Make sure you leave time for the Cistine Chapel and the colloseum. I would recommend the Fawlty Towers hostel. Near the central station, but very clean and friendly.
Then down the coast. Naples is well worth seeing (the home of pizza). A little bit rougher than other Italian cities though, so you might not want to stay there overnight. Especially as Sorrento is so close. This is a touristy town, but still nice to wander through and has some quite good beaches. Also has great limoncello! from here you can easily get to the gorgeous island of Capri (a must-see). You can also take a day trip to see Pompeii (where Mount Vesuvius erupted and petrified a whole city).
From Naples you can then get an overnight ferry down to Sicily. This is a lovely part of Italy. I was only in Sicily for a week, but found loads to do in that time. Agrigento, Catania and Taormina are all worth visiting. You can also take an excursion or hire a car to go to visit Mount Etna. Palermo is the main city on Sicily and well worth visiting. It has a lovely Cathedral and port area. Try as hard as you can to stay at Georgio's Hostel. Georgio is the nicest guy ever, and runs a small hostel with about 5 bedrooms. He makes the best breakfast ever (Canolli, yum!) and takes you anywhere you want to go in his car! He can give advice about anything you need and is excellent value.
So, I hope that gievs you some ideas. Get in touch if you want more info on a particular place!
Have fun, Ruth.