Do you imply that Amtrak might be more advanced than Trenitalia? Frankly, Amtrak is pretty primitive by Italian standards.
If 5 of you are traveling together, then you should buy tickets with the INSIEME ("together") or the FRECCIAFAMILY offer (the latter will be available if there are children under 14).. These offers are available up to 10 days prior to travel.
When you enter the number of passengers (specify the number of adults and children), in your case 5, the INSIEME and/or FRECCIAFAMILY offers will be displayed. Children under 14 travel free in groups of 2 to 5, where there is at least one adult.
Everything should be displayed in your phone (I'm assuming you are the head of the family). As in airplanes, tickets are nominative, so you must specify everybody's name when purchasing. All you need is the PNR, which you could even print on paper, in case you fear your phone might die while you travel.