What is the best way for a family of 5 to travel from Rome to Montalcino (2 nights) then to Vernazza?
Susan,
Probably by car. You don't want a car in Rome and the car is of little use in Vernazza but you really need it for Montalcino.
Pick up car as you leave Rome and dump it at La Spezia.
Getting a family of 5 from Rome to Montalcino and then on to Vernazza using just public transportation would be a bit of a challenge, requiring a combination of trains and buses. Henry's suggestion about a car is the way to go. You'll also need a car if you plan to visit wineries or the nearby towns of Pienza and Montepulciano. I had a car when I stayed in Montalcino and would definitely recommend that option.