In my opinion a car could be useful (and still far from essential) only around Lake Como and - assuming you are not staying in the city centre - Siena. It would be a handicap in the Cinque Terre, which you can only travel between by train or boat, and a major headache in Rome. Personally, I would do at least Como/Milan to Florence and Florence to Rome by train, and take a bus from Florence to Siena because the station is outside the city. I don't know where your base is going to be for the CT but you can certainly reach this by train also from Florence via La Spezia; from Siena, however, you would have to go via Empoli and Pisa, which would take more than 3 hrs.
If you do decide to hire a car for part of your trip, check that the amount of luggage the 5 of you intend to take will fit in the trunk of the car in question (5 large suitcases might not), check the price of fuel, look into parking facilities at the places you will be staying at/visiting, look up how to use toll roads (go to a booth marked "biglietto"!), and above all, check ZTLs in the cities you would be driving to/in, or you could end up paying extra even a year or more later (which also applies if you manage to stray into any bus lanes.
You won't need a car for the CT or for Rome. I'm not sure about the Lakes but having one for Tuscany will be a good idea. Be sure to get an international drivers permit for anyone who might drive and read up on all the traffic rules and regulations.
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