I have a couple of thoughts (beyond "What a great family!"):
- Many, many high-speed trains make the Rome-Florence run every day. If you can pin down the date of that trip way ahead of time, you can buy a non-changeable ticket at a super discount. The trip is actually faster by train than by car. You'll save at least an hour, plus the hassle of renting and returning a car (which people say takes more time in Europe than in the US--I've never done it). And this very forum is littered with posts from people who had a lovely vacation in Italy, then--months after returning home--received one or more very costly tickets for slightly exceeding the speed limit, driving in a non-traffic zone in a city like Florence, driving in a bus lane, etc. Driving isn't hard, but it appears that navigating in the big cities and sticking to the speed limit are challenging.
If you want to see small towns in Tuscany, renting a car (probably in or near Florence) is a good way to do it. Just to move from Rome to Florence, it's really not a good plan.
- Though I've never been to Verona, I understand that it is a lovely city. I'm sure the experience of spending the night there is different from spending the night in Venice. Still, as close together as those two cities are (just over an hour by train), I would use Venice as my base for all four nights. Changing hotels is a hassle and will take more time than expected. Granted, you'll pay substantially more for a hotel in Venice than in Verona, so that's something to consider as well. But spending more time in Venice gives you a bit more flexibility when it comes to planning your sightseeing. If you just love Venice, you can make just the one side-trip, to Verona. If you feel you've seen all the best of Venice and still have a day left (which I think is unlikely), you have two other very interesting cities even closer than Verona--Padua and Vicenza.
I think your basic distribution of days looks good.
It's on transportation that you stand to save the most (in percentage terms) by booking way in advance. You didn't mention trans-Atlantic flights, so I guess you have not arranged those yet. You should start monitoring flights now. Check the "multi-city" [Edited] box and price an open-jaw round-trip from your home airport to Rome and back from Athens to your home airport.
Trains for next summer are not bookable yet, but you can go to TrenItalia and price out your major transportation legs. Look at what each one would cost tomorrow; that's probably the maximum you would pay for that trip if you bought the tickets upon arrival in Italy. Now price the trips well into the future. (I found that at least some prices were loaded for January, but none for late March.) Those prices will give you an idea of the advance-purchase price available if you're OK with a non-changeable, non-refundable ticket. You have to weigh the potential savings against the loss of flexibility that comes from buying your long-distance train tickets well ahead of time.
I wouldn't worry too much about the Venice-Verona leg, because there are fast regional trains that only take an extra 10 minutes or so, and their prices will not increase as you get closer to the date of your trip. You can hang loose on that one, if you want to.
TrenItalia insists that you use the Italian names for the cities, and you may need a specific station name as well.
Roma (Tutte le Stazione) -- that covers all stations
Napoli Centrale (where you'll probably arrive from Rome)
Napoli P.za Garibaldi (from which you'll take the Circumvesuviana to Pompeii)
Firenze S. M. Novella
Venezia S. Lucia
Verona Porta Nuova
For hotels, booking earlier will probably yield places with more charm for the buck (or euro), but you have time to read a lot of online reviews (try booking.com for reviews, and to see hotel locations on a map) and come back here to ask for input.