Although I haven't been to Sicily, I did spend a week or so in southern Italy. The Amalfi coast is simply amazing...I highly recommend checking it out. Pick your method to see it -- bicycle, moped, motorcycle, car, whatever. Of course, you'll have more photo opportunities to pull off and take great shots if you go with two wheels over four.
And of course, southern Italy is where Pompeii and Herculanaeum are, some of the finer ruined cities in the world. I highy recommend staying in Sorrento rather than the urban chaos of Naples. You can get to Pompeii and Herculanaeum from Sorrento via the local Circumvesuvia train line.
Another great thing to see are the temples as Paestum. There are three Greek temples there (southern Italy in ancient times was known as Magna Graecia for its Greek influence). One of them is the most complete Greek temple in the world.
The hilltop towns of southern Italy are scenic, too. There's a lot of other stuff there that I didn't have time to see, so I recommend getting a good guidebook and reading up on stuff, so you don't miss anything that you'd really have liked to see.
Feel free to ask any other questions...
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"Life is not measured in the number of breaths we take, but by the places that take our breath away..."