Rome to Positano - Train or Rent Car?

A car isn't really needed, unless you'd like to have one to visit other places that are not well served by public transport.

From Rome to Positano these are the steps:

• Fast high speed train from Roma Termini to Napoli Centrale (http://www.trenitalia.com/tcom-en)
• Circumvesuviana commuter train from Napoli to Sorrento (http://www.eavsrl.it/web/en)
• Bus from Sorrento to Positano, with the Sorrento-Amalfi Sita Bus (http://www.sitabus.it/wp-content/uploads/2016/06/5070_amalfi-sorrento_dal130616.pdf)

If you want to travel in style you can also use a car service from Naples to Positano (about 100euro instead of the 10+ euro for the commuter train+bus).

Another option is to take the train from Rome all the way to Salerno, basically instead of stopping in Naples you proceed a bit further south to Salerno (use the trenitalia website above for schedule). Then from Salerno you could take a boat (actually a fast hydrofoil) to Positano. This would be a nice alternative that doesn't require as many modes of transportation and gives you a chance to see the coast from the sea. (http://www.positano.com/en/ferry-schedule)

If while in Positano you want to visit Pompeii, the Pompeii site is located along the Circumvesuviana line half way between Naples and Sorrento. You can use the same train to visit there. You can also do so on the way back to Rome (the site has luggage storage).
If you want to visit the island of Capri, there are boats that go from Positano to Capri.
If you want to visit Amalfi and Ravello, just take the same Sita bus. Positano is half way between Sorrento and Amalfi.
So a car isn't essential at all, and actually it adds to the cost a lot, because besides the rental car cost, you need to pay for parking that in Positano (and all over the Amalfi coast) is expensive (over 25 euro a day) and difficult to find at times.
Cars are not allowed in the island of Capri (only residents of the island can)

I rent cars a lot when in the countryside and exploring little villages, but for this trip the car will just be a pain. You are going point to point, so a train or private driver is best. I have driven the Amalfi Coast (white knuckled and don't need to do that one again)... Just take the train and plan for a little romantic picnic and enjoy your ride!

As I've said before in these situations, if Terry Kathryn says no car, that's definitive. She is someone who has stated a strong preference for car travel, so when even she says it's not worth it, heed her advice.

I'll agree with the rest: ditch the car and use the trains/buses.

I'll mention this in case you overlook it on the http://www.eavsrl.it/web/en website. If you can make the times work, there's a seasonal Circumvsuviana CAMPANIA EXPRESS train between Naples and Sorrento which makes fewer stops than the standard commuter train. It's a few euros more but something I'd definitely opt for if possible.

http://www.eavsrl.it/web/en/content/campania-express-2016

We did the fast train to Salerno, then took the ferry into Positano. It's just a 5 minute easy walk from the train station. Beautiful to see the Amalfi Coast from the sea. Just make sure you have a backup plan in case they aren't running due to rough water. We found the bus schedule, and that was out backup.

http://www.travelmar.it/en/

I was in this area last summer at this time. We hired a car and driver to meet our flight in Naples. The driver with Mercedes (common in Italy) was waiting and the plan was to drive us to our villa in Praiano which is just on the other side of Positano. The drive was beautiful, but so glad we were not the ones driving. Very white knuckled drive. Once in out villa we used the Sita bus to go almost everywhere we wanted. It is very inexpensive to go back and forth anywhere on the coast. We took the bus to Positano on two different days, and then took a boat to Capri one day. Another day we took the bus to Sorrento. Traffic is horrible to drive in but the Italians are expert at it. I would NEVER rent car and drive myself. We also called the car and driver to pick up us at our villa and take us for a day in Pompeii. He actually arranged our tour for us in Pompeii. When we had to leave we called the car and driver to pick us up at our villa and take us back to Naples for the flight home. Sadly!!!! If I was going back I would do the car and driver again.

Thank you all for your recommendations!
One more question: any favorites for a private car service in that area, in case we decide to go that route?

Another thought, which I have done.

Train to Napoli, taxi to ferry, and ferry over to Positano. Nice little trip around Capri

There are many plusses to Positano, but you need to understand that it is poorly located for certain daytrips. You should make your departure day plans too. Find out if your hotel has parking before you even think of having a car. BTW, Positano only has one road, with one lane each way going through it. I presume the cheapest hotels are 100 or more STEPS up from the road!

Sorrento is not half as romantic as Positano, but we stayed there for the express purpose of facilitating day trips. We PAID for the romance by booking a cliffside $$$ room with a constant view of Vesuvius, and sometimes Capri. Note that Sorrento has no nice beaches, and Positano only has pebble beaches.

We went to Pompeii (right at the station) one day on the train, and another day to Herculaneum (15-minute walk.) There is no train on the Amalfi Coast, unless you count Salerno.