Publix!!! And if your kids like chicken tenders, be sure to pick some up from the deli there---they are delicious. The Publix by the Wal-mart is a bit larger in size and selection, FWIW.
New this summer: a neat looking gourmet shop and restaurant at Plantation Center in Sea Pines. It's quite large, includes a big wine selection, plus lots and lots of interesting food items for sale---most out on shelves but also some fresh entrees and items in a big deli case. Worth checking out if you like that kind of thing--I know I do! They also have wine tastings on Friday night along with some special activities during the week.