By 10 days, do you include your arrival and departure days? Is there any way you can extend your stay? All three of those locations are so wonderful that it would be great to spend more time. Especially if the 10 days includes your arrival and departure days, you might want to consider limiting your stay to two destinations.
Definitely fly open jaws - on airline websites it's often called "multi-city" - where you fly into one city, and out of another.
Train travel has flatter rates than air, but do vary somewhat. Often "famiglia" fares will be available, which are cheaper, and I think 12 year olds qualify the family for the family rate. Go to the trenitalia.it website and put in some random dates to check fares - the fare structure only lists dates 60 days out, I think, but you can at least get an idea of what the fares would be.
We took our 10yo daughter to Rome and Venice (and a bunch of other places in Italy) last fall, and we paid 199 Euros/night for a triple room in Venice, and 210/Euros/night (190 Euros if we paid cash) for a triple in Rome. Breakfast was included in both places, the locations were extremely convenient, and both were clean and comfortable. You could probably find cheaper places.