I have an aunt and uncle who own a vacation home on Nantucket and visit the island all the time. I'm a little more lukewarm about Nantucket, but considering the dearth of quiet vacation spots in the northeast part of the US, I can certainly understand why people are drawn there.
Nantucket is a relatively small island ringed with lots of sandy beaches. It offers several beaches, all of which had soft sand and were clean. Many of them are also tended by lifeguards, which is very reassuring when you have children with you. The typical beach on Nantucket has parking about 100 yards or so from the water, with a path through the sand to the beach. It can be a bit of a hike to some of the beaches, but nothing horrible unless you're hauling too much stuff. Travel light and you'll be fine. The beaches are very clean, which is remarkable considering how many people flock to them every day.
The interior of Nantucket varies from extremely overcrowded areas packed with vacation homes to more sparsely populated expanses of low vegetation and weather-beaten trees. There can be a lot of traffic on the main roads, but a recently initiated bus system is aimed at curbing that problem a bit. You should be prepared to do a lot of walking and/or bike riding though, because the main town in particular can be unbelievably congested with traffic with no parking anywhere.
"Very nice beaches"
The beaches located on the east side of the island are, as you might expect, quite windy with a high surf. That makes them less suitable for small children, but still quite enjoyable. Surfside Beach is a great example. Kids can certainly have a great time playing in the sand there - You just have to be careful about letting them get too close to the water because of the pounding waves. The beaches on the west side of the island are much calmer and have shallow water, which makes them ideal for children. Jetties Beach is a good example. Ironically, Children's Beach isn't much of a beach at all, so much as a tiny strip of sand that almost seems out of place in the crowded main town. The beach at 'Sconset is a bit more remote and secluded than Surfside and Jetties and makes a great spot if you want a little more privacy.
Although admittedly quaint and scenit, the main town is full of overpriced restaurants and shops where people overpay for everything from hamburgers to jewelry. If you go to Nantucket you should definitely not miss a chance to walk around the town, but it can be kind of hard to imagine what it was like 200 years ago with all the traffic and congestion present now. And if you're brave enough to drive your car down the bumpy cobblestone streets, don't even think about trying to find a parking spot. You're much better off getting downtown on foot, bicycle, or the bus system.