You’ll never run out of things to do with kids in Rome! The Roman Forum is a good place to start – plenty of room to climb and run. At the end of the Forum is the Colosseum, where gladiators killed lions, and if you have enough energy, afterwards you can climb the Palatine Hill to visit the palace of the Roman Emperors there. Really the Palatine is better than the Forum, even – you have it mostly to yourself, and the kids can explore.
Your kids will also love the Golden House of Nero, because it’s mostly underground now and creepy, like cave exploring (if it’s open – it is often closed and sometimes you have to get tickets ahead of time. Ask at your hotel.) Nearby, you can walk to the Forum of Augustus, the Forum of Caesar, and then the Forum of Trajan. Up above the Forum of Trajan, and easy to miss, is Trajan’s Market – the kids will be astounded at the resemblance to a modern mall.
On the other side of town, don’t miss the huge round temple of the Pantheon, or St. Peter’s and the Vatican museum just across the river.
Take another day to ride the bus to the Baths of Caracalla, and then continue on to the Christian catacombs on the edge of town. As a final touch, my kids loved the Capuchin chapel decorated with human bones :)
And… and! you can take the city commuter train to Ostia, and explore an entire ruined city. Kids can climb around on the ruins as much as they like, and there’s a cafeteria too. Stop at the beach on the way back for a perfect day!