Hello Lyangee! It's good to say where you're coming from (If u were from N. America, "short break" not possible!!); by clicking on your name I find you're now in UK. And "spring" covers many months ... if you want beach & warm sea, I think your best time would be mid-May onward. Beach is fine for sunning earlier, but water would be too chilly, for both adults & children. Mid-May is perfect, and the Perfect Island (AND beach) would be NAXOS, and the shallow sandy beach Ag Giorgios (St George) just adjoining the port town. .
Naxos is not a package-tour destination (because its tiny airport doesn't take direct flights), its visitors are independent travelers, both families and "grown-ups." And St. George is not a "family resort" --- in fact (thank God) it is not a "resort" at all, in the UK package-tour sense. The beach is lined with mostly family-run hotels (10-20 rooms) The Scandinavians love this beach & time of year for holidays with pre-schoolers, water is shallow at edge, and warms quicker than elsewhere ... I love sitting on a sunlounger and watching Norwegian toddlers waddling along the water's edge lugging sandpails. Some lodgings that are especially handy for you are Studios Kalergis, Studios Thomais, Sigma Studios, Hotel Asteria. Many have units w. a bedroom plus mainroom (w daybeds) and w kitchenette + balcony, all right on the sands. You can feed the tots on their schedule. Also Low-key tavernas (restaurants) right on the beach. AND when u want to sample the town's eateries, winding lanes, shops, nightlife, it's an easy flat 10-15 min. stroll along the seafront. You can bring a push-chair for the youngest ... let them have ice-cream while you eat dinner, & fall asleep while u linger over coffee. No need to rent a car; if you want to explore the island, the bus network is very good. Don't Delay in visiting Naxos -- Trip Advisor just named it in its "Ten Best in the World" list, and it's being "discovered," alas.
Naxos usually has just 1 flight per day (usually late morning) in modern small jet (35 seats) thus flights fill up months in advance, so knowledgeable folks book waaay ahead. There are also 2 sailings per day (7:25 am, 5:30 PM) on the large modern Blue Star Ferries (1200-1500 psgrs)... these larger ferries are good for younger children, if they are restless u can walk them around corridors, and on deck; reserve "air seats" inside for napping during the 5 hr trip. Many UK travelers seem to find flights that arrive ATH airport in wee hours, 2:00-3:00 ... They take a x96 bus (1 hr ride at those hours) to the Port for the early sailing. When u book ahead, a hotel/pension will have someone meet u at dock w. car or van, and you'd be in your room within 20 minutes of landing.
Here are some photo albums by another Naxos Fan, showing town, beach etc:
A week in Naxos June14 - https://www.flickr.com/photos/stanbr54/sets/72157645113682278/
St George Beach - https://www.flickr.com/photos/stanbr54/sets/72157632113170279/