Rome FCO to Salerno to Amalfi

From FCO take Leonardo Express to Termini
Then fast train to Salerno
Then catch ferry ( or bus) to Amalfi
The Salerno train station is just steps to either ferry or bus to Amalfi, slightly downhill

You would not need to buy ferry or bus tix in advance
Fast train tix purchased in advance will be a big savings but be sure to allow enough time from FCO to Termini or just plan to purchase that ticket when you get there

Adding to Christine's good advice, just be sure to get to Salerno before the last ferry of the day departs for Amalfi. Current schedules are here but do be aware that the ferries don't run between these two locations in the winter season (you don't mention when you're taking this trip):

https://www.sorrentoinsider.com/en/ferry-schedule?from=salerno&to=amalfi#

Thanks Kathy

We quite enjoyed our 2 nights in Salerno so if OP can't make that last ferry it's not a bad plan to spend a night in Salerno!

Well there ya go: bonus valuable advice! 🙂👍

The Frecciarossa train service is much cheaper bought months in advance, but is then not exchangeable or refundable. Plan (... time ... ) your departure from Amalfi before booking the next itinerary item.

Thank you all. We are headed there in early July.