Villa Borghese Gardens for sure hands down. Google it now for pictures. From Roma Termini, Piazza del Cinquecento, buses 360, 910, 223, 92 will get you there in 15 minutes. Wikipedia calls it the third largest park in Rome. So there are plenty of open public spaces to go around.
Also many families lounge on the Spanish Steps. The many outside cafes offer sitting places. Piazza Venezia has park benches around it. The Piazzas already mentioned have sitting places. Yes Rome is busy. It's Rome.
Trevi Fountain has many stairs for sitting. It will reopen in October 2015. But you can go there now and take the catwalk up close to the statues. When are you going? Children have great Fun at Trevi.
How old are your children? It may be wise to push them in kiddie transport carts. Rome requires a lot of walking. Rome is Huge.
You don't have to leave Rome for rest. The bus trip to Ostia Antica is long, it's far, you have to transfer. You have to walk a lot there.
Even in busy Roma Termini, you can go to the huge Chef Express Cafeteria on the Main floor and sit at nice quiet tables for lunch and supper. There are Oases all over Rome.