Rome is a wonderful city and there are a number of areas which would be suitable for family life. Aventino would be particularly suitable for you as it is close to the FAO and most of the international schools. It is also centrally located and served well by public transport links. However, the property prices here are fairly expensive. Alternative options which would suit famillies with children would be Balduina and Monteverde.
Take a look at our page on Areas and Suburbs in Rome for a better overview what each suburb has to offer.
The decision on whether or not to buy a car would ulitmatly depend on where you chose to live. If you chose somewhere central like Aventino - you would not really need a car as you would be better off using the metro to get around. Rome is known to be a very congested and chaotic city - and as a results expats find driving around town too stressful. However, a car could be useful if you were planning on travelling and taking the family out of the city for the weekend.
For more detailed information on purchasing a vehicle I would advise to read the Expat Arrivals Guide to Buying a Car in Italy