Travel by bike in Italy is so nice, but not really easy. It depends which Country are you from!!!
It's not like cycling in Denmark or Holland, for example. Less safety and more hard (due to mountains)
Anyway it could be a great experience. I love cycling with my road bike, and travel too.
Italy is a long Country (2000km) and pretty wide too. If you wish to visit it whole, you need at least 2 months and good legs!
Plan a trip is not too difficult, cos we haven't cycle path. You just need a good map and you'll ride in 'normal' road.
Ride in italian road could be wonderful or awful! Avoid main roads, it is better.
So travelling in small, rural roads will be the best idea. you'll visit nice landscapes almost everywhere and you'll meet nice towns, less expensive, more 'italian'... you'll love that.
Concerning places to visit... too difficult to suggest. too much different places depending on your interest.
Alps in the north, with lakes, tiring ways uphill... good food.
The center, wonderful middleage towns, smooth hills, sea... good food.
South, good wheater even in fall and winter, the best beaches... good food.
and, of course, italian cities, no need presentation.
If you need any help contact me.