Greece seaside towns from Athens

One possibility is to take the metro down to Piraeus, then jump on a ferry for the short trip over to the island of Hydra. The ferry docks at a very pleasant seaside village which, since there are no cars, makes for very pleasant strolling. Lots of attractive waterfront cafes, and if you're especially adventurous you can hike up to an old monastery which offers magnificent views of the Saronic Gulf.
It's an easy day trip from the city.

I would second Hydra. Another close in Saronic island option is Aegina, a place many Athenians escape to for the day.

Glyfada is very close to Athens, a suburb really, and the Hotel Palmyra is right at the water's edge. We stayed there one year for a convention, which was being held at the Convention Centre just across the road. Easy access to Athens. Lots of good restaurants within walking distance.

Instead of Hydra, much closer is island of Aegina -- especially during the week (on weekends it's so convenient Athenians flock there for a getaway). It's only one hour vs 2-3 for Hydra; be sure to take the REGULAR ferrry not a hydrofoil. THe latter are closed-in, no deck or view, the former have big decks to sit & enjoy the whole seascape - giant freighters, tiny sailboats, ferries, cruise ships, waves, gulls - even dolphins if u are lucky! When u arrive u can take a bus (or taxi or rental car) to mid-island hill for FAB ruined temple!! then scramble downhill to a beach town for lunch and a swim, and taxi or bus back to port town ... wander its back streets, have a drink at harbor cafe, ferry back at sunset & quick Metro to central athens. Here's a great photo album that gives u hour-by-hour vision of an Aegina daytrip in June -- https://www.flickr.com/photos/36264706@N03/sets/72157621604646139/detail/ One may I did an overnight on a Monday, solo, and couldn't have enjoyed it more!

By contrast, Hydra is a pretty, picturesque port, you can ride a donkey cart, buy pistachios and then what??? No car or transport to explore further Lots of tours go here, you'll be surrounded by tour groups.. The best houses, views etc are private & hidden -- for the international set.... and you'll have to walk to get there, and peek over a hedge at how the Rich 1% live.

Nafplio is one of Greece's loveliest towns and has a beautiful Old Town along with very nice waterfront, historic sites right in town, great food, reasonable pebble beach, nice walking and just a wonderful place for a day trip. Bear in mind it takes about 2 hours to get there and the same to get back so leave early either by car or take the bus which is a very good deal with clean buses and a nice ride.

If you want to go to a nearby island then Aegina would be a top choice . . . however, you can't get there via train or bus (LOL!)

Having visited both Aegina and Hydra we much preferred Hydra for its laid back ambiance, and if you're indeed looking for something akin to the Amalfi coast then I think you'd prefer it over Aegina as well. The ferry takes about an hour and a half from Piraeus, rather than the 2 to 3 as previously suggested. The charm of the place is that you'll need to do some walking to explore all of the little coves and byways which for us was half the fun. After spending a few days in the frenetic atmosphere of Athens it was nice to take a break in a beautiful sea side setting.
Maybe we got lucky, but during our visit (in September) we didn't see any organized tour groups, nor was the little town clobbered with people. Quite the contrary, compared to Aegina it was pretty quiet, with most of the other people we saw being day trippers like us from the same ferry we rode over on. It was easy to slip into several seaside cafes for a snack or a drink.
Swimming wasn't on our agenda that day, but I do recall that there were some very nice (though rocky) beaches with crystal clear water. Am sure you can inquire locally if you'd like to go for a dip.