It is complicated, but the kids will get some sort of discount on fast Freccia trains, like Venice to Florence.
Children on the fast trains are 4 to 14. On Regionale trains 4 to 12, over that are considered adults.
For the Venice to Florence route, you get slightly cheaper SuperEconomy (advance purchase non-refundable, non-changeable) if you travel at 11:35, the price for 4 adults is 99.60 EUR, but for 2 adults and 2 children is 73.80 EUR. You have to buy these 90 in advance to be sure you can get these prices. Also, there may be a slight fare increase after December 12.
And for routes that have Regionale connections the pricing is will be a combination. The 14 year-old will be a child on the Freccia, but an adult on the Regionale.
For Florence to Lucca, you could take a Regionale train instead of the bus. So it would be 3 adults at 7.50 EUR each, and one child at 3.80 EUR, so total would be 26.30 EUR.
You can get all the prices at http://www.trenitalia.com/tcom-en
As was already pointed out, you must use the Italian City names, Venezia, not Venice. FI.
There is also another program called Bimbi Gratis (children ride free). Prices are not quite as good as SuperEconomy tickets, but they have fewer restrictions as far as changing the time and travel date..