We will be traveling with 4 adults and 2 little boys under the age of 2. If the boys are sitting on our laps will we need to purchase tickets for them?
Children under 4 are free but no seat is provided. Of course if the train is not full, no one is likely to object to using an empty one. First class on a Freccia you are likely to have to keep them on laps; possibly an issue in 2nd class as well on the Freccias since seats are reserved. In the regionale trains, you'll probably have more freedom if there are empty seats.
In Italy (different in other countries) ALL trains except the regional trains require a reservation and with that reservation comes a seat assignment. As stated, children under don't need a ticket but they do not have a seat either except for the regional trains where no seats are reserved. Depending on where and when you are going would decide the best train for you.