On a budget, Villa Borghese Gardens still earns its price.
Free (boat/bike rentals extra)
A green reprieve from the stone and crowds: rent a rowboat on the little lake, hire a bike, and walk to the Pincio terrace for a postcard view over Piazza del Popolo. It's huge and a bit unmanicured in spots, but it's the easiest place in central Rome to let kids run.