FCO to Amalfi

We went a few years ago by taking the train from Rome- Termini to Salerno, then a boat to Amalfi. There is also a bus from Salerno to Amalfi. The boat ride was very nice, scenic.

You can check with alilaura.it for boat transit from Naples to Sorrento, Ischia, Capri and Amalfi Coast. From FCO, get to Roma Termini and take the Freccia to Naples (fast and frequent). The bus from Sorrento is a 'tourist' thing. Don't even consider the circumvesuviana train from Naples to Sorrento - a sure vacation spoiler. The boat ride (ok, you're inside the boat not on the deck) is far more interesting and relaxing after the long flight from home.

You could also consider renting a car and drive, if you don't mind doing so while jet lagged.
Another option could be a private car transfer with driver. This option is expensive (about 350 €) however depending on how many people are in your party, it might be financially advantageous as well as comfortable.
If neither alternative appeals to you, then the train+boat suggestion above (via Salerno) is probably the best option.

This link has a complete list of options you can take from FCO in Rome with different routes and is quite intuitive to follow:

http://www.ravello.com/how-to-get-to-ravello/

The guide is to Ravello, true, but Ravello is only a bus-ride away (~7 km) from Amalfi. So, all routes lead to Amalfi and then as last and only option to reach Ravello, you take the bus.