We took our 5-month old son to Rome. A bit nerve-racking in the planning phase, worried about the long flight from the USA and jet lag. So if you're planning a trip to Rome with kids, know that it CAN be done. Just a few things to share to make your life easier.
First, ditch the stroller. Rome if full of cobblestone streets and our stroller was useless. Instead bring a baby carrier (like a sling or an Ergo). It will make your life much, much easier.
Second, if you can, try to wake up late / stay up late. It helps with the time-clock adjustment -- and best of all it means you can live like a Roman. We kept our son up until 10pm most nights, to help with the time adjustment. And somehow it worked (at least the first few nights, until the jet lag caught up with him).
Third, if you rent a car (like we did) bring your own car seat. The rental agencies claim you can use one of theirs, but the quality sucks.
Fourth, don't try to do too much. I wasn't used to traveling with kids - I was much more of a 'do it all' kind of person. Traveling with a child taught me patience. It's a lot more fun to take it slow with a happy family, than to cram too much into an itinerary.