Sorrento to Amalfi Mid-June

I’m big on the ferry. The views from the water are the best. Pulling into the dock in Positano might have been the best part of Positano. Picking up the bus from Amalfi is pretty easy, doing this on your own is very viable. You’d miss out on walking around Positano, but most people would agree it’s not that interesting. Very touristy as you’d expect, at least in the lower part of town. Strictly from a scenery perspective, which is 90% of the coast appeal, I’d probably go this route.

On the other hand, a good private driver can put together an itinerary that you can’t do on your own. With a family of four, it’s not a bad way to go.

We loved Sorrento. We did a day trip to Amalfi with Rick Steves suggested SharedTours.com

Check them out online - we were very pleased.

We used Aldo Limos to tour the AC coast, last May, 2023, but if I had a do -over, I would have done the trip via the water. Aldo's was very professional and a great service, but the AC trip is almost nine hours, and you're only out of the car about 4.5 hrs, because of the traffic congestion. And Positano and Amalfi were packed shoulder to shoulder, and this was May. (Folks with car sickness may not enjoy the winding roads.)
Another lower cost option is Mondo shared van tours of the AC Coast.
The ferries are weather dependent.
The infrastructure in the area has been sadly neglected, and traffic is awful, with public transportation unable to meet the high demand. I was not prepared for the congestion.
However, you have the tools to be prepared for all this and I wish you a wonderful family trip. Safe travels!

It is absolutely reasonable to note how attractive the ferry is. But I question how realistic it is to assume that you will have free choice of any walk-up transit option or another, during the peak season in a frightfully crowded area. You need a back-up plan for whatever you choose - busses too full to take more passengers, ferries cancelled for high waves in the gulf, always brutal traffic on a two-lane mountain road, weak infrastructure in a low-budget part of the country, last-year's construction on the Positano docks, whatever.

Just one example: Every day we walked by the Sorrento Circumvesuviana station in MAY, the line for the bus to Positano/Amalfi was longer that would fit on the NEXT half-hourly bus. Rick (our host here) has reported that you should visit Positano first on an AC day, because the "bus home to Sorrento" from Amalfi may be so full that it does not even try to stop in Positano on the way back.

"We have 1 day" may be a little vague. Are you sleeping in Sorrento, Salerno, Naples, or are you talking about a day trip from Rome, or something over-ambitious like that? You'll get better answers if you are more specific.

https://community.ricksteves.com/travel-forum/italy/amalfi-coast-day-trip

With 1 day and OK with spending the money, I'd recommend a driver. Traffic won't be fun, but those twisty roads were part of the experience for me. The ferry is a good way to get you to a starting point but at some point you'll need to rely on ground transportation as well.

No matter your method, it will be slow going. We were part of a small group tour in April 2022 and of a 9 hour day starting in Sorrento and going to Positano, Amalfi, and Ravello, 4.5 hours were in the vehicle. On another day we took the ferry to Amalfi with intentions on catching the Travelmar ferry to other locations, but the schedules never worked out and we ended up taking the bus to Minori and then back to Amalfi from Maiori. This was also a 9-10 hour day with a lot of time sucked up with transportation.

We are staying in capri for 2 nights and sorrento for 3 nites before going to Rome