Amalfi to anywhere is bus/taxi/driver or ferry to a larger destination - usually Sorrento or Salerno. And from there you can get the train to various locations - probably through Naples to Rome and then onto Civitavecchia.
TrenItalia is the state train agency that operates most trains and .Italo is a private company that operates only fast trains like those between large cities. I would always buy the tickets directly from these companies versus a third party supplier like Omni for clarity and the most options. Both companies have apps that I would recommend getting and using is you are buying tickets with them.
Using TrenItalia allows you to book all your tickets as opposed to be combining them yourself. Local tickets are always the same price and can be bought as you need them but you can get discounted (sometimes very discounted) tickets at the cost of flexibility on fast tickets with both fast train companies the earlier you buy. If you are sure of your schedule then you can book those tickets earlier at a discount.
There is a whole industry around transportation from Rome to the port of Civitavecchia if you search for it but you can also simply buy train tickets and make your own way.
Hope that helps, have a great trip,
=Tod
