I would suggest you take the train from Rome to Salerno, a nice ride on one of the fast trains. In Salerno, you have several options, either rent a car or take the bus. We did this trip in 2016, took the train from Rome to Salerno and then rented a car across the street from the Salerno station. It is a lovely drive to Minori although some may say it is a challenging drive. If you are worried about the narrow roads and curves and traffic, the bus is a good option.
"should we take trains and ferry or bus?"
Yes, on outward journey.
This is the way:
Salerno-Minori by boat (by far preferable):
(search for "Salerno Concordia": this pier is at walking distance from Salerno railway station)
Salerno-Minori by bus:
As for the return trip to Naples Airport, there are few direct bus rides from Salerno:
(from Minori to Salerno: see above)
Otherwise, a direct private transport is required (ask your hotel).
A car is not a reasonable option along the Amalfi Coast unless your accommodations come with parking. Even then, if you move the car to go elsewhere along the coast you'll face parking issues.
I think the easiest and most comfortable travel route is the high speed train to Salerno followed by the ferry to Minori. The ferry dock at Piazza Concordia is just a couple of short blocks from the train station, an easy walk.
You can see the ferry schedule here: https://www.travelmar.it/it/orari
For the return you can take the ferry back to Salerno, from there you can take the train to Naples followed by the airport bus or you can take the train to Rome and change there for the airport train.
Naples to FCO bus: www.fiumicinoexpress.com/orari-partenze/…
There is a Pintour minibus which will take you directly to Naples airport from Minori for 15 euro per person. You can book in advance. The schedule is here: https:/…orario-2018-pintourbus.com_.pdf. We travelled with someone who used it both to and from the airport in May and it was quick and efficient.