My family and I want to go from home to Paris, somewhere on the French Riveria, (open to suggestions), Rome and Venice, in no particular order. We have about 12 days. I'd like advice on method of travel (plane, train or automobile?) in between cities while keeping cost in mind. We have two children, ages 13 and 16 also.
If you are heading as far as Nice in the French Riviera, you have both train and plane options. It's quite a train ride from Nice Ville to either Venice or Rome but the train is an option. For the plane, there are budget airline flights from Nice to Venice (Air Berlin), Milan (Air Berlin) and Rome-FCO (Air Berlin, Meridiana/Eurofly or Blu-Express). If you book really early, fares can be quite cheap. Just be careful to read the carry-on and luggage size and weight restrictions on the website.
Jeanne,
Given the circumstances you listed, I'd probably arrange the trip along these lines:
To begin with, book open-jaw flights: inbound to Paris and outbound from Rome. Keep in mind that you'll lose the first day in flight times.
From Paris to Nice, travel by TGV would be a good choice. These depart from Paris Lyon, arriving at Nice Ville. The shortest travel time I could find was a direct train at 5H:29M. Of course, with ANY of the "fast trains", reservations are compulsory.
For Nice to Venice, a budget flight would be the quickest mode of travel as going by train will take a minimum of 7H:38M. However, you'll have to balance that against the cost. I had a brief look at Air Berlin fares and the cheapest I could find was €129 per person. You may be able to get children's fares either on rail or air, so the cost will be one factor to consider.
For Venice to Rome, the EuroStar Italia is your best bet. Travel time from Venezia Santa Lucia to Roma Termini is as short as 4H:27M (again, reservations compulsory).
My preference would be to place Rome at the end of the trip, as it's a bit easier to get to the airport than in Venice. Of course, the availability of flights to & from your area will have a bearing on that.
Happy travels!