I will be traveling from Varenna to the Genoa area for three days then to Lucca for one day and night and then to Rome for two days and then fly back to the States. I would be using the car on to get from city to city. I am thinking that I will save time by driving, don't know about the cost. Does driving seem like a good choice? Thanks for you help.
It is really personal and trade-offs that you are willing to accept. There is more to driving than just point to point. Parking is always a big issue. We are just off of nine days driving in the country side of Tuscany. One day we spent over two hours looking for a parking space and one other day it was over an hour. Big advantage of train is city center to city center. Use a car if there are no other ways to conveniently get there.
For the car part: Figure twenty-five cents a mile. If it's five hundred miles, that's $125. Figure twenty-five bucks a day for parking. If you drop the car when you get to Rome, that's $100. A wild guess at the tolls is $50. A cheapo car plus the insurance that you credit card won't cover is probably about #350. (Weak guess, probably low, I rent elsewwhere and drive into Italy and I also rent for longer terms so the daily price drops.) If the five hundred miles is close, that's about eight hours of road time. I've driven hundreds of thousands of miles in Europe and maybe thirty thousand in Italy. I always have a care, I've never not had a car, I'll never not have a car, I step off the plane into a car and step out of the car at the airport. Were I doing what you propose, I'd be hanging around train stations. The traid dudes can give you the good news. I've no idea how to compute it.
I always check out the cost of the train vs a car for my trips in Europe. I find that, not counting parking, a car would have cost 2-3 times what I actually spend on the train. A car will usually take a little less time, if you don't count the time at rental offices and looking for parking. With a car, the driver, at least, has to watch the road. On the train, you time is entirely your own, to watch the scenery, read, keep a journal, sleep, whatever.
The Route Planner at http://viamichelin.com allows you to plot out a multi-leg (+ Step) journey by auto anywhere in Europe or the UK. You can select many filtering options, and the results will tell you the amount needed for tolls, gasoline, and in a recent query I did it even listed the amount for 'road tax'. It may not prove to be precisely 100% accurate when the rubber actually meets the road, but it's a good tool for getting an idea of the costs. If you plot out an auto journey, then also try to research the price for train tickets between cities along a similar route, the results might be enlightening and help you make a decision whether to rent for all or part of your trip, or concentrate on using public transportation. Good luck!
Unless your trip includes substantial amount of countryside and small village driving, renting a car is rarely a good choice. Varenna can be easily reached by train and lake Como is better enjoyed on the ferries. I'm not sure what you intend for Genoa area. In Genoa itself a car is not needed. At the Cinque Terre it's not needed and many other coastal towns are easily reached by train and parking is scarse everywhere. The train from Milan to Genoa is faster too. You will need a car only if you visit some small towns on the Ligurian Alps. In Lucca itself a car is not needed. Actually cars are not allowed inside the historical center (inside the city walls) unless you are a resident with permit. Same with Rome where cars aren't allowed within most of the city center inside the Aurelian walls. The only advantage of a car is flexibility and the ability of better transporting the luggage. It's a convenience that comes at a price: money and stress for the driver.
albert, Unless you're planning to visit small hilltowns or other locations that aren't well served by public transit, well planned travel by train would be easier, more efficient, probably cheaper (especially if you pass through ZTL areas) and definitely less of a hassle. All of the places you mentioned have excellent train service. For the longest travel segment, Lucca to Rome, the majority of the trip will be via Freccia high speed train which runs at up to 300 kmH. There's no way you'll be able to match that in a car. Depending on which train you choose, travel time on that leg will be ~3H:35M. Happy travels!