Take a fast train from Rome to Naples. Go down into the station basement and take a local train to Sorrento, then a bus to Amalfi and another bus to Furore. Alternatively, have a car take you from Naples station to Furore for around €100.
For the best prices on the fast train, book now. Tickets for 12 September went on sale on 12 May, and the cheapest tickets have probably sold out by now. You can get train times and prices and make bookings at www.trenitalia.com clicking the flag at top right to change to English and using the Italian versions of city names (Roma Termini, Napoli Centrale), or its competitor on major routes, www.italotreno.it
Personally, I would take the train to Salerno instead of Naples, then have a car service take you to Furore OR take the ferry from Salerno to Amalfi and get a taxi from there to Furore; the car service from Salerno probably about 100 euros - from Naples around 110 plus euros -
Ferry schedule Salerno to Amalfi - coopsantandrea.com/it/…
If you prefer car service - go to Naples and get car service from there; if you want to reduce spending - go to Salerno, ferry then taxi to Furore - skip the 60 plus minute train ride Naples to Sorrento - then awful bus ride (crowded, late, etc)
Thank you for quick responses. This is really helpful. I have booked fast train from Rome to Salerno, will likely take ferry or bus to Amalfi and maybe a taxi for the last part of the trip. We would like to experience the crazy bus ride to Amalfi but ferry also sounds interesting. Is the train station too far from the ferry terminal? How do we get there? Or is it easier to get to SITA bus?