Not that I have to be, but I am a very value-driven traveler.. Some would say cheap, but I have found in Italy--and by the way not in the US--that one can exist very well in certain parts of Italy for not a lot of outlay, and still have most if not all of the creature comforts.
First, there's the matter of airfare. Starting about 6 months out, look at Kayak or Google flights. Overall, if you can get where you want to go, there and back for under $1000/pp out the door (reference point for me from Chicago) after seat selection, baggage and all the nickel/dime crap that airlines hit you for these days, consider it a job well done. Sure, you can do a 18-hour layover in Istanbul for $550 R/T, but does your family want to do that?! Hard to get good, cheap air unless you have points, period.
Contrary to popular opinion, I would consider both Florence & Rome places you can stretch a Euro. Yes, find the best AirBnB in a central area for your family, and find out what the average rental prices are in the area, and that can change by neighborhood. Be forewarned, if an apartment seems too good to be true...it usually is. When looking, you can decide whether you need that 2nd bathroom, or 3rd bedroom. Read the reviews. Make sure there's a washer, so no trips to a lavenderia. Don't know the size of your family, but a 2 BR, 2 bath shoulder season can be gotten for around 200-250 Euro/night. But that's for all of you. And now you can set up shop. Shop at the bakeries, fruit markets, don't eat out more than once a day, if that, although the right restaurant in either Florence or Rome can be gotten for 20 Euro/pp or under. In both cities, there is such competition with so many restaurants that price-wise they have to stay in line. Also, unbelievably great sandwich shops abound--Nerbone in Florence, Forno in Rome---3 Euro per sandwich or pizza slice.
Walk everywhere--in Florence & Rome you can do that, easily. That way you save Uber or bus fares. And there are plenty of free or very cheap museums in both places. City to city, 2nd class train fares bought 3-4 months in advance will get you 60-70% discounts, so utilize that for sure.
Of course, in southern Italy you can find lodging and meals even cheaper. We've stayed in Salerno--in Campania just off the Amalfi Coast--and found a wonderful hotel right across from the train station for 100 Euro/night and pizzas for under 10 Euro. Once you get onto the A.C., prices jump 50%. But to see the main stuff--why you go to Italy--do your due diligence researching in both Florence & Rome and find good deals. It can be done.
Enjoy your planning!