Ciao Phil!
Your kids will have a great time here in Rome. As far as surrounding cities... Ostia Antica is great! We took middle and high schol kids there and they wandered until exhaustion, then we sat in the amphitheatre and had a picnic lunch. Afterwards they all invaded the gelataria there and went home pretty happy (and tired)!
I've also done trips with kids to Ovierto. The Duomo to them was "okay" but they loved the cave tours - especially the story of the pigeons( won't spoil it for you!). Granted, most of the kids living here have probably seen 10,000 "sites."
Tivoli is pretty in the Spring. Kids we went with loved the villa with it's fountains.
The beach is always interesting. Okay, our first field trip with "teenagers" we had to take the boys aside and talk about the "topless" factor. No problem they said. Sure enough, once they got there, it was, in LOUD English...., "Hey come look at this one over here!" AARRRGHHH! Set back Italian-American relations for years probably. Most beaches near Rome are private and you can reach them by metro/bus/LIDO train. You pay a fee per person for the day. There are a few FREE public beaches but you have to know where they are! For most flexibility, go by car!
There is a "monster" park north of Rome that many kids like...I've not been yet. Bomarzo Monster Park is located 60 miles north of Rome near Viterbo, off Highway A1; or travel by train to Attigliano and then take a local bus.
For water fun... you've got these two:
Acqua piper - Via Maremmana Inferiore - +39 0774-326-538 -
This is an aquatic park with pools with waves, water slides, pools for toddlers, electronic games, a restaurant, and a picnic area.
Hydromania - Vicolo Casal Lumbroso - +39 06-66-92844 - This large water park with an Olympic-sized pool, slides, water massage, kamikaze, etc. Exit 33 off the GRA.
There are also plenty of lakes around Rome. You'll have no trouble keeping them busy! Good Luck -
Ciao,
Ron