I recommend Aegina, stayed there 2 nights once just on impulse, because couldn't get my ferry to Sifnos. It's got a lot more to it than Hydra -- the latter is promoted by Steves as "no-cars" island but in point of fact, Aegina has VERY little car traffic at all, off season. It has an interesting port town, lots of tavernas & shops. Since it's so convenient to Athens, it's a favorite getaway for Athenians so something is always going on.
It's the closest island; there are 2 types of vessels (1) Hydrofoil 45 minutes, do NOT recommend because all closed in, now view, and (2) regular ferry, 1 hour, thse have sheltered open decks where you can sit out, and enjoy views as u leave piraeus, and all the water traffic -- from freighters, cruise ships, ferries, to small private craft ... on a sunny windy day in April maybe even sailboats. And YES, there are frequent ferries so you could readily catch a 7 AM ferry, arrive 8 AM, get an X93 bus (€5) that would take you straight to the airport in about 70 minutes max.
I'd advise staying in the port town ... but you can take a bus to center of Island, to Temple of Aphaia, the bst-preserved ancient temple on any Greek island (although the sculptures were ripped off in 1830 or so by a couple of BRit Oxford types & sold to a museum, grrrr). From the temple, theres a long road (and a quick downhill shortcut path) to the beach at Ag. Marina & you can lunch there & catch a bus back. Here's a wonderful photo album of a daytrip to Aegina http://www.santorini-hotels.info/santorini-island-santorini-map/santorini-firostefani-map.htm The Official Island website has this section on highlights, v. informative ... http://www.aeginagreece.com/aegina/pages/sights/index.html A lot of locals there actually commute to the mainland daily for jobs .... and if you have a sunset drink in a harborside cafe, there are lots of these socializing ... and (as you'll discover) almost all Greeks under 40 are fairly fluent in English, I always find sociable types at harborsides.
Booking.com lists Aegina Town accommodations, I suggest looking fairly close to harbor, perhaps Electra Pension, nothing is that noisy in April ... places with pools out on a beach are a longer walk to tavernas, cafes etc.