Your largest single cost will likely be your plane tickets. Here are some considerations you'll need to take into account before deciding how much you'll spend. First is that if your route contains an intermediate stop in the US you will have to clear customs there on your way back into the country. This will make your total flight time a bit longer and will require you to gather you bags and re-check them to your final destination. I don't like this, so I choose a direct flight to a European destination, even if that destination isn't in Italy. Multi-hop flights are exhausting and generally cheaper--your choice. To find which flights are available, you can use Rome to Rio, entering your home airport and the destination city. It will return all the options it can find, along with a range of prices. You an even book from Rome to Rio.
If you have questions about a flight on a certain airline, I find that you can get quite good service by finding the FaceBook page of the airline and asking your question.
Car or Train? If you're planning to get out into the Italian countryside you'll want a car, although maybe not a Ferrari. You can rent a car for short trips. We find that AutoEurope offers the best customer service along with reasonable prices from your choice of providers. For a two week trip starting a Fiumicino Airport in Rome, prices start at around $300 for an October, 2017 vacation. For longer trips you may wish to lease a car. This is a special French program in which you get a brand new car of your choosing, great insurance and don't have to return the car full of fuel. Our leased Peugeot cost us around $1600 for eight weeks. There's no added cost for a second driver. Check out Peugeot Open Europe Leasing.
So how does the cost of a car stack up against the cost of a train? The more people you can cram into your car the more budget-friendly it becomes, although you'll have to watch closely the number of suitcases you bring. The train is very convenient for trips that take you to the heart of big cities like Rome and streetless ones like Venice. New, faster trains can save time but cost more than the regional trains. Remember also that if you want to go a decent speed between places in Italy, you'll use the autostrada, which will add more than you think to the cost of a trip. And finally, in 2017 the price of unleaded gasoline in Italy is about 1.5 euro per liter. That's over twice what it is in the US. Some of the sting of that price is lessened by the fact that European cars get significantly better mileage than US cars. Diesel fuel is cheaper in Italy as well.
We find that the cost of an Italian regional train for two people is about the same cost as driving if you use the autostrada. The car will cost more if you pay for parking.
Italian Trains are an efficient way to move between large cities. If you dread the thought of driving, parking, and paying autostrada tolls, taking the train is a relaxing way to see Italy, and despite what people say, you can get to smaller places on the train. Rome to Florence, then Venice and back to Rome should cost $127-190 per person depending upon the speed of train and class (2017). The trip from Rome to Venice takes 3 hours and 36 minutes.
Find out more: Italy Rail Map | Book Tickets
Italian hotel prices are a bit above the European average. You can find a decent, mid-class (3 star and above) hotel in most large and medium size cities in Italy for $80-110. In smaller places and in the south you'll find less expensive hotels. Hostel rooms are even less. Read more: Hotels in Italy.
Check Prices: Hostels | Hotels
Food is one of the reasons many folks go to Italy. The standards are quite high and food is generally no more expensive than it is in the US. If you drink wine, your restaurant check will be quite a bit less than it would in the US because wine is considered a normal drink rather than a way for a restaurant to make lots of profit. Expect to pay $40 a person from a good meal in a city, less in rural areas.
Now, as to saving money:
Above are the tools for you to figure out your travel budget. But you want a stock answer, don't you? Well, if you're frugal (you don't want to pay for frippery, but you'll shell out for decent food, wine, and attraction tickets when those things enhance your vacation) then you might spend $150 per day. Add your own airfare because I don't know where you're starting.
And have a great trip to Italy, whatever your travel budget.