Sorrento to Amalfi in early May

Take the bus one way, ferry the other
For the bus you will want to line up early for first bus of day if you can
They do fill up

The ride is nausea inducing so ferry both ways is also an option

There is also the Red City Sightseeing bus
Don’t know when your trip is, if it will be in service

Here is a partial answer to your question:
https://community.ricksteves.com/travel-forum/italy/amalfi-coast-day-trip

When in Sorrento I noticed that the prices on the Mondo Taxi signs were not less than we paid other vendors. But a full-day car hire is certainly a negotiable product.

It is better to give the month of the year when you ask a question like this, because crowding in the Sorrento and AC area is severe in high season. In the last week of May, every day we went downtown in Sorrento, we passed the Circumvesuviana station, where the SITA bus to Amalfi [town] stops. Every single time, there were more than 55 people in the line, which means some people were going to wait for two busses before they got on, and the chance to get a seat on the window side was slim.

Rick has said in some books that if you are going to both Positano and Amalfi in the same day, it is better to go to Positano first, because the return bus from Amalfi may be so full that it doesn't stop at the Positano stop at all! No personal experience on that.

Edit: Pre-Covid I think there was an intermediate price tourist large-bus company (transportation only, not "tour") that made less frequent trips along part of the AC. You might find the name by searching this forum, top center.

I believe Tim is thinking of City Sightseeing. Unfortunately, it appears that it is not currently operating buses in the Amalfi area--perhaps not anywhere in Italy at all.

Hi Carly, my friend and I were considering the same shared-ride idea. Mondo tours also offers a private driver RT to AC or you can request a private tour guide and driver. We elected to hire the guide and driver for the day, since we only have 3 nights in Sorrento. Mondos is also providing transportation back to Rome for our return flight. They were very helpful in getting our brief stay organized. There are less expensive forms of travel around AC and we felt this is the safest and most convenient. Perhaps you could send an email to Vincenzo @ Mondo Tours stating your plan . If your 'shared plan' sign-up has less than the required travelers, he might provide a separate RT driver. It's worth a try.

When I visited the coast, I took a late morning bus from Sorrento to Amalfi. The line was long so I had to wait for the next one. Luckily, I was able to snatch two seats on the right facing the driver. The coastal road is one-way that travels west to east. I stayed on the bus until it stopped in Amalfi and was sick from nausea so wear motion sickness bracelets on each wrist. You may also want to ask your physician for a prescription. The views were spectacular but you have to tolerate gas fumes and constant curves.
Another idea to help avoid motion sickness is to get off at Positano then hop back on to get to Amalfi then take a direct ferry back to Sorrento (1h 15m).

I don't have personal experience, but note that the Positano harbor can only fit smaller ferries than some other AC ports. I presume that means a rockier ride.