I know Rome very well, since I lived there for 6 years and worked for a local tour operator.

For the appartment, I would stay at the hotel for three nights, you are in holidays after all and you deserve to be served and just have a great tzime... but it's my personal opinion.
I agree with the other writers about the Express by Holiday Inn San Giovanni, it's really nice and stylish!!
I do not agree at all on the Jolly Hotel Vittorio Veneto, not a nice location and old style. Don't go there if you are staying at the Sorrento Hilton after that!! It's another world!

But you could also check the website www.booking.com, there you can book hotels (not only in Italy) at good prices and WITHOUT TO PAY ANYTHING IN ADVANCE!!! They'll have offers for Rome in summer for sure, cause it's low season. If you want an advice on an hotel you find, write me: I know most of Rome hotels.

For families, Accor chain is very convenient, they have a nice hotel close to the Colosseum (Mercure Delta Colosseo), childrem under 16 don't pay but they have to share the parent's room.

AS Roma and Lazio training locations are not open to visits, nor does the Olimpico stadium. It is a pity actually, I tried to organize tours there but it seems to be impossible... they don't understand business)).

The best way to go from Rome to Sorrento is actually rent a car. It's a nice journey and you can organize yourself as you want (important with children along...). Check with Herz or Europcar websites, best fares are to be found online, you can probably pick up the car in Rome and drop it off in Sorrento.

Best wishes for your trip and just ask me if you need more advice...