Book as early in advance as you can. Same with hotels. Train tickets you don't need to worry about all that much but look at getting an Trenitalia pass early on too. Train travel is way more civilized - and a million times safer than driving in Italy. Rome is seriously dangerous with how they drive.
Huh, if it were me I'd skip Como and Lucerne. They're pretty but there isn't a whole lot to do in those places besides gawk at how pretty they are, eat and shop. There isn't a whole lot in the way of culture or learning. My suggestion would be to retool your trip. Fly to Rome, spend at least 2 full days there - go the Vatican museums - see the Sistine Chapel! - and St, Peters Basilica on day one - lineups can be long for St. Peter's but it's worth it. It's one of the most amazing man made structures on the planet. You will not regret it. I tell everyone going to Rome, if nothing else, go to St. Peters! See the Pantheon and the Colosseum, or skip the latter and go to the The Galleria Borghese instead - incredible art museum!
Four days in Florence is acceptable. You'll only scratch the surface but you can see a lot in that time. The Uffizi should be first on the list. Go to the Bargello National Museum there. It's a real gem that's overlooked by most tourists so it's never crowded. The Basilica of San Marco is seriously impressive if you want to see just what kind money and power the Medici really had. The Pitti Palace has some great, more modern art as well as the Boboli gardens which would be in full bloom while you're there.
Then spend 2 days in Milan. Maybe even fly out of Turin. The more major places - not Como/Lucerne - will provide much more options for keeping your kids entertained, as well giving them some important culture early on.