I have the perfect island for a family like yours; I know because I've sat on the beach there, many a May, watching happy toddlers staggering around with sand pails, while relaxed Scandinavian parents watch. NAXOS. and the beach is Ag. Giorgios....Sandy & shallow, with many budget hotels/apts right on the sands. A web of small lanes, no thru traffic. Beach is Directly adjacent to the Port Town of Naxos, so u can walk to port side cafes, shops etc in 10 minutes along level waterside, pushing tots in strollers, have your dinner there while kiddies have snacks, bottles, fall asleep in chairs while u enjoy sunset serenely. Also tavernas right at Beach. No need for car except if you want to rent 1 for a day to zoom around the island. Here's a wonderful photo album of Naxos hi-lights: http://www.flickr.com/photos/stanbr54/sets/72157634605629689/ This map shows port, town, and location of beach, plus some of the interesting sights http://www.naxos-greece.net/villages/TOPOGRAFIKO.HTML
Other Plusses: Many moderate-price lodgings have family-friendly "studio units/apts" (1 or 2BR, balcony, main room w. 2 daybeds, kitchenette & table/chairs); this enables u to feed kiddies when THEY are hungry, not in a restaurant. Plenty of small groceries for food, diapers, etc. AND if u can book a place with Sand-level patio units, saves on all those up/down steps... and while a tot naps, u can sun right outside. For right on sands, check STudios Thomais, Studios Kalergis, Sun Beach, Hotel Akrogiali. FOr places 20-70 meters off sands -- Hotel Elizabeth, Sigma Studios, Villa Naxia, Hotel Panormos. But don't delay!! Families book these early!
As photos show, Naxos Town itself is FASCINATING; history, ruins, best museum in Cyclades, I never tire of exploring it. AND a final plus for your island-hopping wish -- Naxos (unlike most islands) offers day-trip options. 2 companies have dedicated vessels going 2x a week to Santorini & 2x to Delos/Mykonos (I like the MV Alexandros best; larger boat, smoother ride). I could see taking the kids to Santorini, the Delos/Mykonos could be too much.
Naxos is not plagued by cruise ship crowds or jumbo-jet throngs. But it does take planning to get a family there quickly. Only one flight (45 minutes) from ATH daily around mid-day, or a 5-hour ferry ride. Flights fill up VERY EARLY because plane (modern; new) only holds 35-40 passengers. I advise flying TO island, ferry back (get "reserved economy" seats on large modern Blue Star Ferry, & kiddies can walk the decks then snooze in your lap). IF you choose Naxos, (1) FIRST book Aegean/Olympic ATH-Naxos flight! (2) Then get flight Neth-ATH early enough for connection. (3) book lodgings ... there are a lot more rooms than plane seats.
Believe me when I say, Naxos is the ideal answer for your family ... I've sent MANY young families there and they all loved it. I remember what it was like to plan a vacation with pre-schoolers; the right location is everything. Oh, and did i mention that the prices are half that of the "famous" islands ... you can dine for €10-12 each, many family units are €60-70 or less in May. GO!