Easiest way to get from Rome to Positano

Train to Salerno, then boat to Positano (weather permitting).

By the train(trenitalia.com) to Naples or Sorrento then by the local bus ,that said, during the high season (which runs from April through October), this bus is packed. So packed, in fact, that people often wind up standing in the aisle, holding on for dear life, swaying with each curve. It’s a grueling way to travel and it can go on for an hour and a half, depending on the destination. Worse still, many of the buses have very poor/non-existent air-conditioning, which can be tough to handle when summer temperatures get up to 100°F. One other way to get around is the ferry. It’s often faster (one of the most frequent Amalfi to Salerno ferries takes about 40 minutes, opposed to nearly 1.5 hours by bus), less crowded, and gives you beautiful views of the towns from the water. One downside, though, is that the ferries are more expensive than the bus (that 40-minute Amalfi-Salerno ferry cost €7, opposed to €3.60 for the 90-minute bus card).If you travel with kids take a look to special tour for kids in Rome4kidstours or Europe4kidstour with English private driver.

You asked for the easiest way, well, that would be to hire a car/van for your group and drive there. If you want to hit a ferry simply take the car to the Naples marina a take the ferry from there. By the way, it is uphill from the Positano dock to any hotel, and I mean uphill.

Here is a website to consider: https://www.my-limo.com/?rlimo Cost is around $500 euro.

5 adults, 4 children?! Yikes.

I would originally say ferry Naples/Positano--if it's April through October, you don't specify--but it could be a P.I.T.A. to get that many people from the Naples train station to the port, and you had better have a Plan B if the weather knocks out the ferry.

Book in advance through Trenitalia (so you get assigned seats) a Freccia train Roma Termini to Napoli Centrale. Kids will love it. Then, also in advance, book a big van--and make clear you'll have 9 people and plenty of bags I imagine--to meet you when you get off the train and drive you Naples to Positano.

Yes, it'll be pricey.

Enjoy your planning!

Hand's down, I'd go with Roberto's choice:
"Train to Salerno, then boat to Positano (weather permitting)"

Here are current ferry schedules:
http://www.positano.com/en/ferry-schedule

In the event of lousy weather (boats can't dock in Positano if the sea is too rough):
There is a bus from Roma to Sorrento that runs once or twice a day depending on the date. It takes approx. 3 hours, 45 minutes - 4 hours, and departs from the Tiburtina Autostazione. The site is only in Italian but with some help from google translate, you could probably figure it out...or sweet-talk Roberto into helping. :O)

A one-way ticket looks to be 19.00 €
http://www.marozzivt.it/home

From Sorrento, I'd consider getting a driver to Positano. There are ferries from Sorrento but that's a moot detail if they can't dock.

(oops: edited a typo)

In the multiple posts above you have every option possible.
The best to me also sounds like the Salerno ferry but as Kathy mentions anytime there are rough seas nothing docks at Positano so that is out and not based on weather but sea conditions so tough to predict depending on time of year.

Also, major difference on where you staying, considering you will have many bags. If staying closer to the dock arriving by ferry much better, if staying near the road arriving by road (bus or private car) is better.
Would not underestimate how miserable the walk from the ferry to the road would be carrying 10 bags, though some hotels may be able to arrange for a fee a porter service.

Sounds like you have been there before and had a bad experience on the SITA bus already?
I took the bus Kathy mentions and it was very easy though not at all on-time. The bus stopped multiple unscheduled times, was very laid back way to get to Sorrento which we enjoyed more than the train.
With a big group it would be much better IMO as long as you are not in a hurry and realize it is the slower option. Just easier with no changes.
From the Sorrento bus station you could arrange for a car to drive you to Positano or take a cab, there will be cabs waiting there, you may need a couple of vans!