The Rome Transportation system makes it easy to get around the city with lots of options if you don't feel like walking.
I'd have to say walking is always my first choice, because there is so much to see on foot, and also because I love the exercise! And, Rome is pretty compact and you can walk easily from monument to monument, plaza to plaza. But it's not always feasible to walk due to weather, tiredness, trying to fit in a lot of stuff in a day, hurt knees etc.
Other than walking, here are the best ways to get around Rome:
Rome's public transportation is run by ATAC (Azienda Tramvie ed Autobus del Comune di Roma). Unfortunately their website does not work well in English. The ATAC system includes:
Rome Metro system - the Rome underground is an easy, quick way to get around parts of Rome...the parts not full of underground Roman ruins!
Rome bus system - Rome has an extensive bus system, allowing you to get almost anywhere in Rome, even to many parts outside Rome. It's the slowest way to go, but it can be an adventure and/or scenic if the bus isn't too crowded.
Rome Tram system - There are a limited number of trams in Rome, all part of the ATAC Rome public transportation system. The trams are all in strategic spots, connecting to either a metro or bus stop, making it very easy to get quickly to locations the metro cannot always go, such as the auditorium or MAXXI museum.
Taxi - you may decide to just take a taxi in Rome, but you should know the rules first!
Driving - Sorry but I must strongly urge you not to use a car as a way to get around Rome.
If you are renting a car and driving in or out of Rome, fine. But there is no need for a car in Rome, it's not easy or cheap to park it, and you risk getting slapped with hefty fines for driving in restricted areas that you may not be able to recognize.
Bicycle - I have a bike but I don't ride it to get around Rome. Unfortunately there are too many wacky drivers and I know of too many (bad) accidents.
But going on a leisurely bike ride in Rome is lovely. It's a great option for a ride in a park, or along the Tiber, or the area around the Coliseum where there is no traffic.
There are a very few other ways to get around Rome, such as: