First time to Italy and skipping Rome, Florence and Venice? Just do the countryside?
A private driver for that distance will cost you more than a transatlantic plane ticket.
I would rent two cars to go to Praiano too. You should rent two small economy cars because they are cheaper to rent than one van, give you more flexibility if you want to do things separately, they are easier to negotiate the narrow streets on the Amalfi coast and Tuscan towns. Also, I don't know where you are coming from before Praiano (Rome?) but getting to Praiano on public transportation is a hassle (one fast train to Naples, then a commuter train to Sorrento then a bus to Praiano. The only draw back is the cost of parking at the Amalfi coast (about 25 euro a day). If you don't want to rent a car for Praiano, I would at least certainly rent a car in Sorrento to drive to Cortona. You will need a car in Cortona to do day trips in Tuscany so you might as well do it from Sorrento and save yourself a bus, two train changes and the train fare for six. Just send two drivers to get the car, drive back to Praiano to pick up the rest of the party and the luggage, then drive to Tuscany. And why would you want to return it in Arezzo? (Which is possible, if you want, since in Arezzo there is Hertz, Europcar and Avis for sure). Just return the cars when you arrive to Florence, where you don't need a car. Avoid having the rental car while in big cities (Rome, Florence). But a car is more convenient for countryside and small towns' visiting. Rent when leaving Rome and return as soon as you arrive in Florence.