Every war has theaters of operation. In World War II, for example, the theaters were Europe and the Pacific. For you and your war against grocery costs, your theaters will be home and the supermarket. In both places, you will find unique ways to save money. Let's start with the home.
Your very first step should be to make a grocery budget so that you won't overspend at the supermarket. Once that is completed, look at things you can do to cut the amount of groceries you need to buy. Growing some of your own produce is one way to avoid plunking down extra change at the local grocery store (though you'll find these savings after paying the upfront costs of things like tools, plants, seeds and soil enhancements). Not only will this provide your family with plenty of fresh veggies, fruit and herbs in the summer, but it will also give you a chance to can and freeze food for use throughout the year.
And speaking of freezing, storing food is another way to save money on groceries. If you have room in your freezer -- or if you invest in a separate stand-alone freezer (assuming you have the space) you can make the food you buy last longer. Anything from bread to milk to meat can be frozen for later use. And if you vacuum seal and freeze the leftovers of foods you make at home -- things like soups and casseroles -- you're less likely to waste them.
Now, your next challenge is the supermarket.