Trenitalia has a version in English as well. The only thing you have to remember is to enter the to/from destinations in their Italian names - although if you start typing in the boxes, a drop-down appears to choose the one you want. Rome's central station is Roma Termini; Naples is Napoli Centrale; Salerno is just Salerno, etc.
So you'd want a train from Roma Centrale to Salerno:
http://www.trenitalia.com/cms/v/index.jsp?vgnextoid=4ddd1a035296f310VgnVCM1000005817f90aRCRD
Ferry times from there to Amalfi are here:
http://www.coopsantandrea.com/en/servizidilinea/servizi_b.asp?id=1
Alicost also has a coupe of ferry runs:
http://www.alicost.it/?page_id=202&lang=en
If for some reason you can't use the ferries (wrong time of year/bad weather) you can take a SITA bus from Salerno:
http://www.sitasudtrasporti.it/archivio/Download/corse/sitasud/Campania/Salerno/3b678683-ef88-41b8-af55-b6baa88fe17c_QUADROXV_2015_ESTATE_B.pdf/0
Pompeii on the way to Naples is a little tougher. I'd like to send you on a ferry to Sorrento but can't find service from Amalfi before 11:35 AM (arriving Sorrento 12:50) and from there you'd need to take go to the train station and take the local Circumvesuviana to Pompeii. That train will not be found on the Trenitalia site: I've included the schedule near the bottom of this post.
http://www.alicost.it/?page_id=202&lang=en
To leave earlier, you could take the SITA bus to Sorrento, and the train from there:
http://www.sitasudtrasporti.it/archivio/Download/corse/sitasud/Campania/Salerno/29b6e52b-5251-48a7-bca2-68048bbf2b19_5070_ESTATE.pdf/0
The Circumvesuviana schedule is below: you want to get off at Pompei S. Villa Misteri - which is right in front of the entrance to the ruins. There is a left-luggage office near that entrance.
http://www.eavsrl.it/web/sites/default/files/allegati/Orario%20Napoli%20Sorrento%20con%20Pozzano%20e%20Scraio%20e%20viceversa.pdf
From there, you get back on the same train to Napoli Garibaldi station. Tickets for that train are inexpensive, as are tickets for the bus. The ferries will be more expensive.