Not local, but frequent visitor.
For a quieter/smaller beach I'd recommend Ouaisne. It has a small kiosk & toilets, and parking very close.
My other favourite beach is Greve de Lecq, but there is reduced parking currently so you may need to get there promptly to park rather than rocking up at 11:30.
Another less busy beach is near Green island.
A walk out to Corbiere lighthouse across the causeway is fun, especially if you get there as the tide is going out
5 mile road as mentioned above (St Ouen's bay) has wonderful splashing up waves of a high tide near the Watersplash bar which have always been a favourite with our children (and us).
The zoo is pricey, but you could easily spend the day there. There are very good keeper talks which are worth going to, especially the gorillas as it gets them all visible. The enclosures are good for animals, but that does mean you can't always see everything first time.
We have often taken out Jersey Heritage membership (which may work out cheaper than their offers they to for visitors), but that depends on your budget and interest. But it does mean you can come and go without then worrying about the prices. (However mine didn't like spending hours on beaches so we needed to do other stuff). 2 castles, Hamptonne farm, interactive Maritime Museum are all very good, (plus some other places).
I wouldn't do the underground hospital (now Jersey War Tunnels) with children of your age.