Reputable companies throughout the city offer rickshaws, bikes, or overboards for rent. Be advised that the new mayor just banned unlicensed rickshaws from operating, so always choose agencies and companies that offer these services, don’t just drop Euro notes to the first unknown approaching you near a tourist site!
Also: there’s a variety of “moving tours” available, from vespa rides of the city to segway tours: google them up and choose carefully, and consider that most alleys in the center of the city are closed off to mopeds, or have cobblestones and could be unconfortable for you and your kids.
You could also go the “carrozzelle” way: a more expensive tour of the main monuments done on a traditional small horse-carriage, the way most Romans used to go about before the cars invaded the Eternal City! Other options include renting bikes and horses - we reccomend both for exploring greener areas such as some of the parks in Rome or the beautiful Appian Way, just South of the city walls: an ancient Roman road, still in use today!