Not knowing your family's interests, I suggest you get Rick Steves' Italy guidebook and study the sections on the Amalfi Coast (what town?) and Rome. Only you can determine if a day trip to Pompeii is your family's cup of tea, or if you want to buy skip-the-line tickets for the Colosseo before you leave home. Do you want to go to Galleria Borghese? You need to buy tickets ASAP if you do. Vatican? Book a tour with Walks of Italy. Lots of planning to do before you fly!
FYI - you do not need a vehicle for any of this, and you certainly do not want to drive in Roma. In fact a vehicle is a hassle for both the Amalfi Coast and Roma. Go local and take the trains.