Rome to Venice stopovers with 16 year old girl

Renting a car between Rome and Florence to visit Tuscan/Umbrian towns makes sense.
I don't see the need to have a car after that, actually a car would be a hassle.
In your shoes I'd do:
Rome: visit with no car
Rent a car in Rome or Orvieto then head north to Tuscany.
Visit Tuscany with car for a few days.
Arrive in Florence, return car upon arrival, stay in Florence a few days.
Take train north to Venice (maybe stop in Bologna along the way if you wish). Stay in Venice for a few nights.
Take train from Venice to Lavagna (via Milan).
Lavagna, is a pretty busy beach town that in summer has quite a bit of traffic and a big harbor/marina. I don't know what exactly you intend to do there, but if it's for beach and maybe visiting other towns in the area, the train is very convenient and it's right across the street from the main beaches. Unless you have specific needs for a car, I don't think a car is necessary for staying in Lavagna. You'll be walking to the beach and taking the train to places around.

I think Roberto has the best plan with the least amount of hassle and the beauty of having a car in the area where it will be the most fun... the countryside of Tuscany. There are any number of wonderful village that you can stay in and still explore the surrounding area by car. Just be aware of the towns that have the zones that you are not allowed to drive in. Once you decide on which towns you are going to you can investigate that.

You can go to Booking.com and see tons of hotels in Venice and Florence in a price range that you feel comfortable with. It's so easy to compare hotels this way.

I use them frequently.
For the car you can go to Priceline and put in your pickup and dropoff city and get lots of options as well. Most cars will be a manual unless you specify and automatic, and the cost will be more.

We made arrangements for my husband to fly out of Venice...so the trip is Rome, then rent a car and drive to Venice, stopping along the way somewhere. And then Venice for three nights and then a train to Florence for three nights. And then going to Lavagna. So what I would really like is wonderful stopping place (or two of them) between Rome and Venice for us to stay overnight. I was thinking that it might be a good idea to drive part way the first night, and then keep going toward Venice the second day and stay somewhere else. That way we would always be moving in the direction and getting closer to Venice to break up the trip. Or am I crazy to try to fit in the two nights somewhere between Rome and Venice. Would it be better to just stay an extra day in Rome and an extra day in Venice and forget the in between places? That way we could take a train from Rome to Venice. But I had really wanted to take my daughter to see some of the most awesome hill towns.

Train. Rome-Orvieto. Spend 4-5 hours explore the town or spend the night. One of the best hill towns.
Rent car, drive to Montepulciano then on to Verona or Pescheria del Garda
Spend night.

Next day train to Venice.

Train. Venice-Florence.

Train bus combo to Lavagna.
From Florence.

That sounds like a terrific plan Gerri. I we take a train to Orvieto, what do we do with our luggage while we are exploring the town. Are there large lockers for big suitcases at the train station? And where do we rent a car in Orvieto?

Also, do you know what kind of car (economy or compact) will fit three of us and three large suitcases?

The trains are so convenient, the only way I would recommend a car is if that is the only way you can see what you want to see. You don't want a car in Rome and you don't want a car in Venice, so don't get one in between! I would recommend Orvieto for a night and Verona for a night. You may even be able to see a performance in the amphitheatre there. We stayed at a RS hotel in Verona and it was perfect and budget friendly.
Do you have stuff you must bring that will prevent you from travelling light? You shouldn't have 3 large suitcases for 3 people. You may regret lugging the large bags around, but you won't regret wearing the same pair of pants a couple times.

I didn't look for luggage storage in Orvieto, but I don't remember seeing lockers in the Italian train or bus stations I went through last year. There was often a left-luggage (staffed) office, however, and occasionally a nearby business offers the service. Note that station left-luggage offices may close at meal times, so check that carefully when you're dropping off bags. There would normally be a sign, indicating the hours of operation.

It can be difficult to pin down information about storage possibilities outside train and bus stations from afar, but the situation in Orvieto comes up pretty frequently on travel discussion boards, and I found this, which sounds sort of encouraging. (Read to the end.)

You should use Google to get addresses and then map the location of possible storage points. When you arrive in Orvieto by train and are headed for the old town, you walk across the street to get the funicular. It's possible that unofficial luggage-storage location(s) would be down at train-station level. You'd want to know that before getting on the funicular with suitcases. I think the car-rental location must surely be down on the lower level as well. Orvieto's historic district has limited vehicular traffic.

The walk from the top of the funicular to the core of the old town, while quite attractive, is not short, and it's slightly uphill. I don't think it would be much fun with a heavy rolling bag. In fact, I know it wouldn't be, because I rolled a too-heavy bag about 1/3 of the way toward the cathedral myself. Completely doable, but not terribly pleasant.