Rome is a wonderful place to visit with kids, especially as simply walking the streets is a sightseeing experience: amazing art and architecture is all around, without any line-ups or admission charges. Visitors who have the time can read up on the layers of history and be richly rewarded (and there's an app for that), but it's also possible to simply stroll and be delighted.
Walking, Resting, Restrooms
If you're going to be doing a lot of walking with kids, several issues come up.
We all want to avoid the scenario of over-tired kids melting down... Most families will be visiting in summer, and Rome gets very hot; in fact, the city empties in August, when Romans head to the beach or mountains for holidays.
With older kids, the fuel to keep on strollin' in Rome is gelato - ice cream. Our general policy toward sugary treats gets left at home when in Italy and when the kids got tired, we took a gelato break. There are countless ice cream stores in Rome -- Click for a photo of the beautiful displays typical of an Italian gelateria (ice cream shop) and read tips about getting the finest.
Rome is full of stairs, which can make stroller use less than ideal with kids who aren't walking yet. See advantages and disadvantages of stroller vs. baby carrier at the Rome for Children blog. Parents of wee kids may want to bring both along.
Parents of preschoolers might consider bringing a light-weight umbrella stroller so that your child can ride when he/she gets tired. When you encounter stairs, the child can get out and walk.
Evening and Night Are Your Friends
Do as the Romans do, and rest indoors during the hottest part of the day. Then enjoy walking to Rome's famous piazzas and fountains in the cool of evening or after dark. The streets will be full of families with little kids, at 10 p.m., 11 p.m....
Families will find plenty of places to sit down and take a rest, either joining other tourists lounging on the Spanish Steps or on the public seating by the Trevi Fountain. There's not much shade in these places, though. Take a break at one of the countless outdoor trattorias and cafes that serve sandwiches and snacks. (A "trattoria" is less formal than a restaurant.) Be prepared to pay a small extra fee whenever you sit down at a table.
With kids along, it's especially important to not end up in a long line-up for a museum or other sightseeing attraction. About.com's Italy Travel site has tips on avoiding lineups; for example, visitors can make use of several types of passes.
Take the chance to use the toilet whenever you stop for a meal at a trattoria. If however, your child needs a restroom immediately after you've just left the place -- surprising how that happens -- fortunately Italians are very indulgent toward children, and you'll probably be treated helpfully if you enter a trattoria with a small child in desperate need of the "WC". ("WC" stands for Water Closet and is the common sign for a restroom.) Otherwise just buy a drink or snack, so that you're a paying customer.
Rome does have public washrooms, but they may be hard to find and reportedly some are not facilities you'd want your child to use. The better maintained public washrooms will typically have an attendant who expects a small fee, so keep some change handy.
Read many details about Toilets in Rome at the Lonely Planet blog.
Unexpected Tip: Families may discover a new love for MacDonalds, in Rome: over twenty are dotted around the Eternal City, and offer air-conditioned comfort, washrooms, and familiar low-cost food.
Using Public Transport
If you're game to do as the locals do, take advantage of public buses and the Rome metro. Visitors can buy passes for unlimited rides for one-day, three days, a week, or a month. Note that these passes and even single tickets cannot be bought on the buses; you need to buy a ticket or pass first. They're available at tobacconist kiosks, vending machines at metro stations and major bus stops, and in some bars. Some attraction passes include public transport tickets too. Read details about getting around Rome by bus. Buses can be crowded, and you'll need to move forward with purpose to get on the bus; don't expect an orderly line-up.
Finally, here's good news for travelers, especially those visiting in hot summer months: free, cold water is available at many fountains places in Rome. (Download a map.) These fountains are called "nasoni" and were first installed in 1874: read more and see a photo of what you're looking for.