Southern Italy~Beautiful Historical Beach Towns via Train/Bus

So I have been to beaches in Salerno, and all the way down in Amantea, and I am not sure that there will be many (if any) "white sandy beaches" like you are looking for on the western side (I don't know much about the east coast). I just checked a few pictures of various towns between Salerno and Amantea and at the NICEST looking ones, the beach would be a bit darker than a very shelly Atlantic coast of Florida. More like grey sand? Is that ok? I just checked Paestum and while the town isn't very historical, the Greek Ruins nearby are VERY historical, and I looked at pictures and the sand is kinda white/tan-ish? They do have private beach club places, but its nothing like a bustling resort area, nor a bustling town. You're in luck that there is a train that goes more or less down the coast to take you to many places. The Santa Maria di Castellabate and Agropoli area (Agropoli by a train, Santa Maria by a further bus) is very quaint and old feeling, but they are tan sand places.

You could probably spend a week somewhere along the coast between Naples and Santa Maria di Castellabate doing random things, but it might be easier with a car the farther you go past Salerno.

The most "developed" resort areas I have experienced don't lend themselves to having "town centers" to walk through repeatedly, and they are more like resort after resort after resort.

If you wanted an easy relaxing beach day, even though I don't like Positano, its very straightforward and RIGHT there in front of the town.

The Adriatic side of southern Italy has history with many of the towns in Puglia founded by the Greeks (although little generally remains except the street plan) and then colonized by the Romans. There is a direct train from Rome to Bari and local trains/buses along the coast. Trani has a lot of history from the Middle Ages as well. Vieste has nice beaches a short distance below town, perhaps reachable by bus. All along Puglia the towns we've visited generally don't have big beaches themselves although there are a number of beautiful beaches a short distance away and there is a bus network you can research online (look for info on traveling in Puglia by public transit) to see if it gets you there.

The west coast of Puglia should be suitable, although it will be a challenge without a car. Porto Cesareo is very nice,
https://slowtravelitalyspain.blogspot.com/2023/11/puglia-in-september-2023-porto-cesareo_26.html
but Gallipoli may be easier by public transport. The sand on beaches in Vieste is golden not white and it's a very long and windy road by public transport. Trani is a fantastic place for other reasons, its beach is not its best point. Best beaches in Italy are in Sardinia, south of Olbia for example, but once again not easy without a car. Or in Sicily places like San Vito lo Capo (fly into Trapani from Rome then bus) or Marina di Ragusa, may be doable by public transport, fly into Catania from Rome.
https://slowtravelitalyspain.blogspot.com/2019/02/west-coast-sicily-day-9-san-vito-lo.html

How ‘bout a ferry?

Ischia is a 1 hour ferry ride away from Naples. X nights in Naples and 7-x on Ischia?

There’s a sandy beach in Ischia Porto and bus service all around the island to other areas with other beaches. The sand may not be white, but you might be pleased anyway.

oooooo Sardegna! That would be perfect for what you are looking for minus the car :(