In Italy (and in Europe in general) there are several ways you can travel and visit other cities and countries, not only by car. And usually they are also very cheap.
Car sharing is a very good option if you are looking to save money and to make friends/connections (not always good 😊), but you have to fit in other people’s schedule (not always possible) and, sometimes, it is not very comfortable.
Buses are ok and cheap, but they are not very comfortable and not really fast. Still a good option for short trips.
About the trains, they can be divided into two main categories: high velocity trains and the standard trains. The first ones can reach 300 km/h and they can take you to Rome in about two hours: they are a little bit more expensive than the standard trains but if you book them in advance they can be extremely cheap (http://www.trenitalia.com/tcom/Le-Frecce, https://www.italotreno.it/it). The other peculiarity is that they usually have just one or two stops, not more. Because of that, if you are planning a trip with several stops for visiting different cities, you may want to consider the other type of trains. The standard trains, called “Intercity” (http://www.trenitalia.com/tcom/Altri-treni), are usually cheaper, slower and less comfortable but, as I said before, still a very good option if you are not in a rush and you are planning to visit all of Italy, not only the big cities (something that I warmly recommend 😉).
Flying in Europe is very cheap thanks to low cost companies (i.e. Ryanair, EasyJet, etc.). They usually have very strict rules about baggage policy, check-in and payment procedures (be sure to take a look to them before purchasing), but they are the best option for traveling within the continent. Tip: plan and book a lot in advance for very good deals!
And, as they always say in this case, enjoy your staying!! 😊
N.B. The bike is also an option but it is not really a big thing here in Italy, so I suggest to be very very careful with that and to plan it in all the details. Here you can find some ideas https://www.bikeitalia.it/itinerari-cicloturismo-italia/, and I am sure there will be some more. To be honest, this not even something I am in too much but I can help you to find them.