Check the ads before you go to the store. Modify your meal plan and list to take advantage of sale items. Leave the kids at home with your spouse or a neighbor. Most of us don't have the heart to tell our little angel that she can't have those special frosted cookies in the bakery section. Stores have been known to put tempting treats within a child's reach.
Larger supermarkets are generally cheaper.
That's due to volume discounts, but check the ads. You might find that it's worthwhile doing part of your shopping at one store and finishing up at another. Nothing says that you have to use the same store week after week. It's a good idea to scout out other stores periodically. Look up and down.