Would I take the Trenititalia from La Spezia Centrale (Cinque Terra) to Rome?
Are these National or Regional trains?
A bit of terminology. Trenitalia is the Italian National Rail company. Website: http://www.trenitalia.com/tcom-en
They operate nearly all the trains, though there are some local operators.
Trains are of two types: Long distance/high speed trains and "Regionalle", both are run by Trenitalia.
For long distance/high speed trains tickets are specific to a train, they have date, time, train number, coach and seat number on them. The tickets are only valid on that train (like air tickets are), and are considerably cheaper if bought in advance.
For Regionalle trains, tickets are open, you can use them on any train; but if you buy them at a station you must stamp them before getting on the train to stop you using them twice.
If you are going from (say) Monterosso al Mare to Roma, that is what you look up on the Trenitalia website. It will calculate the quickest route, which will involve changing trains, and a mixture of Regionalle and high speed trains. But you don't need to know the route in advance, it will tell you which trains and where to change. It will sell you a ticket that will cover all the trains, end-to-end.
Just go to the Trenitalia site, punch in where (and when) you want to go, and get a "print-at-home" ticket.