The First way to save money is comparing the prices of the 2 competing railway companies: trenitalia.com/en.html and italotreno.it/en.
Italotreno runs only high speed trains between Venice and Salerno, stopping at Bologna, Florence, Naples and Rome along the way. Italotreno's discount codes are a real bargain, just subscribe their newsletter.
Second: be aware that a child can travel with any type of discounted ticket, you can always choose the cheapest option. For example, If 4 SuperEconomy tickets are cheaper than the Bimbi Gratis offer on that day, you are free to buy 4 SuperEconomy tickets.
Finally, since you are travelling up and down the boot: learn the correct spelling of the places you are visiting, find them on a map and check the prices of open jaw air tickets. I'd fly into Venice, out of Rome and I'd forget Sorrento (that, as a side note, is served by trains not covered by any pass).
Note that Cinque Terre is the name of an area made of five (Cinque in Italian) villages. You'll have to pick a village where to stay and put that village's name on trenitalia.com