Depending where you'll be staying in Rome, consider a daytrip to Tivoli (Villa D'Este, specifically, but not Hadrian's Villa: the former is easily reachable from the train station at Tivoli but the latter needs a bus connection from the town and it would be tiring, although wonderful). Also, Lake Bracciano and the Odescalchi Castle there: it's a short train ride, 1h10 on average, and the ancient town at the base of the castle is wonderful. Good restaurants, too! Another one could be to Ostia Antica - there's a light rail-sorta train departing every fifteen minutes or so from the terminal at piazzale Ostiense (you need to google "trenino Roma - Ostia Lido") and get off at "Ostia Antica": it's a Roman port beautifully preserved, the experience is similar to visiting Pompeii, only 30 minutes south of Rome!
As for strollers, you could consider renting one when you get in and use that while on vacation. I know of a few places that offer that kind of service, actually!