I'm older and my kids are all grown, but when they were younger, we too were on a limited budget. What I did then, and still do, is to shop the weekly grocery ads. Whatever is on sale that week is what is on the menu. I go to one store for most things, and another one very close by only for their specials.
Also,it takes more time, but if you avoid most processed and all pre-prepared foods, you will avoid the preservatives and additives. Buy things that are as near to their natural state as possible. They are usually found on the outer edges of the store--fresh fruit and vegetables, milk, cheese, meats,yogurt, eggs, milk, etc. On the inner shelves are dry beans, rice, pasta, canned vegetables, & cereals such as oatmeal and cream of wheat.All this is available in any regular supermarket.
It's true that there are almost never coupons for these items, but it's really cheaper in the long run than buying already prepared foods.
However, you have to spend some time cooking. The only way to make sure of what is in your food is to make it yourself. I always made nearly everything from scratch, including breads, desserts, cookies, spaghetti sauce,casseroles & other main dishes.
The other thing to do is read the labels on everything. They sneak high fructose corn syrup into a surprising number of things that you would never suspect. But, you also have to consider how much is in the food, and how often/how much you consume. For example, ketchup has high fructose corn syrup. Now I'm not going to make my own ketchup! But I seldom use it, so I don't worry about the little bit that I might take in. If it's listed as the first or second ingredient, then it becomes a problem.
I personally don't believe that organic is necessarily better tasting or even better for you, but if you can afford to pay extra for it, that's your decision. I even read somewhere that organic chicken is even more likely to have salmonella than regular supermarket chicken because of the organic ones not receiving the antibiotics.
You might also want to check around for local farmer's markets, farm stands that sell directly to the public, or even the classifieds for people selling produce.
Or you could start a garden and grow your own! The kids will enjoy watching it grow and picking their own vegetables, and there's nothing better than garden tomatoes. It's also a good family activity.