Aegina is the best answer! Here's a splendid photo-sequence of a daytrip there in June - http://www.flickr.com/photos/36264706@N03/sets/72157621604646139/detail/
Just be sure to choose (as shown) a Conventional ferry sailing, NOT a hydrofoil... the latter has you pent up in a cabin with no view, whereas conventional (and large)ferry has open-deck for the 1-hr "mini-cruise" where u can sit & enjoy the seascape -- huge freighters & cruise ships, 3-mast schooners, tiny sloops. The evening before you can stop at any ticket agency by your hotel, find out the sailing schedule, & buy ticket for minimal €1 fee ... if the big ferry sailing is too early for u, you can get a hydrofoil outbound and just make sure to get the big ferry back... I think last 1 is about 6:15, so u can have a sundowner on deck, & get back for dinner in Athens. When u arrive @Aegina, check sailings & buy ticket as u leave pier. BTW, Aegina is a favorite weekend getaway for Athenians, so you're best off going M- F if possible... fewer visitors then.
SIGHTSEEING -- You can now rent e-bikes, as well as renting a car for a day or half-day... options for exploring
• Before leaving port town - quick stop rilght by pier for v. small ruins area (only 1 column - great foto op w sea backdrop!) and small cute museum - my fave exhibit items = huge mansize amphora jugs!
• Monastery ? Mainly for flowers
• Aphaia = best preserved ancient temple on any isle; fab 360° view of Saronic bay
• Beaches -- video shows Ag Marina... nice but narrow sand... also look at Marathona - bring suits for midday dip! lots of taverna lunch options.
• Late afternoon -- back in port town, fun to stroll back streets, mainly for locals, I love browsing lttle groceries & housewares stores for Greece brands/items. When u have an icecream or a glass in seafront cafe, most around u will be Greek-speakers.
METRO -- You can now train to Piraeus from TWO Central Ath stations - Monastiraki AND Syntagma... takes about 30 mins. Only caution - be aware that pickpockets target this run... Keep your ID, phone, wallet, cc's etc SAFE -- i.e. not in pockets or backpacks. I use neck-pouch under a shirt, or my day-bag crosswise in front with hand covering zip. (NOTE: I read a few yrs back that most who get "picked" are men who insist on keeping stuff in pockets "I'd KNOW if someone got in my pocket!" No u won't).