You will get many replies indicating that you simply do not have enough time, but I think your plan is fine. Not sure what is on your list of sights or what your interests are, but here are two wide ranging plans:
Start in Sicily working your way North, spending at least two weeks in the region from Sicily to Naples. This will give you an overview of Sicily and decent time in Naples and the Amalfi Coast, but not much in between. From there, move North, Maybe some time in Rome, but maybe pick a spot in Umbria/Tuscany/Florence or Bologna for your month. From there, you can explore much of Northern Italy as Day or overnight trips, being only a couple hours from about everything.
The other option would be to reverse it, but in that case I would devote most of a Month getting to Sicily, winding up with 3 weeks to chill and daytrip.
Do not feel you need to see everything, my first time in Rome was 3 nights, first time in Venice one night, first time in Florence..a day trip, did I see everything...No...did I have a great time and see the highlights, you bet, did I go back, yeah, several times.