Aegina (40mins or 60) is closer than Hydra 2 hrs... People keep raving about Hydra has no cars, but for a day-trip, Aegina has VERY little traffic but handy to have vehicles around. One can take a bus to Island center to see stunning Temple of Aphaia atop a hill (360° view of Saronic bay), then walk downhill to Ag Marina? beach for swim & lunch .... if bus schedule works, take bus back to port town, otherwise taxi. (get a taxi driver's card at the port, to call if u need it). Sunset at port & dinner is great, then ferry back by starlight. THis photo album https://www.flickr.com/photos/36264706@N03/sets/72157621604646139/detail/ shows step by step a day trip in June. NOTE: since Aegina is so popular as getaway for Athenians, best to go M-F, not weekends. I was there in early june on T-W, not crowded.
Another nice beach, Marathona, has a cute cafe for lunch, and more beach room
- https://www.aeginagreece.com/aegina-island/directory/aegina-listing/restaurants-cafes-bars/beach-bar/la-palma-marathonas/
Here's the key: at least outbound, take Conventional FErry, NOT the more-frequent Hydrofoils (latter has no outer deck, windows are salt-frosted no view). With conventional ferry, u sit on deck & enjoy seascape, everything from freighter to huge cruise ships, to schooners, or fishing boats... gulls, even maybe a dolphin ... and enjoy approach to attractive port. If u return after dark, u could take a hydrofoil for convenience. I can't seem to access the usual ferry sites tonight, but I know there's a conventional ferry at about ? 8;30. you can stop into a ticket agency near your central Athens hotel and ask for timetable for NONhydrofoils.
Just take Green line Metro (30 mins) from cental athens to Piraeus (WATCH for PICKPOCKETS; don't carry valuables, etc, just a small zip-coinpurse with €€ for the day) ... buy ferry tickets at a portside agency.