we save a number of ways. first, get yourself or your family a share/membership in a good co-op. in addition to your discount, there you will also meet others who may know how to save using the resources available where you shop.
if there's a public market where you are, they're always cheap. sometimes you can find organic stuff, but you have to ask. most of the stuff at our public market is not organic, though, and if you have chemical sensitivities to toluene & other chemicals commonly used in pesticides it can be really unpleasant, because you have no way of knowing what has been sprayed on the stuff.
look into community supported agriculture. we just found out about CSA where we are. saves lots of money on organic produce, & sometimes you can get eggs, dairy, or meat too. $112 gets us a share, which means we help with a distribution shift & get 16 lbs. organic produce 6 times between now & spring.
go vegetarian, if you think your family will let you get away with it. with really stubborn meat-eaters, if you serve favorite dishes that just don't happen to not have any meat in them (lentil soup, home-baked mac-n-cheese, red beans & rice etc.) rather than "vegetarian" dishes, most of the time they won't notice or care, especially if your motive is saving money. ignore the over-priced meat analogs. vegetarianism really is cheaper! simple grains & legumes are the cheapest foods in the world! in a pinch i can feed a family of 4 (well, one of us only takes breastmilk) for $100 or less for 2 weeks of groceries.
use your freezer. freezing does damage some of the nutrients in foods, but making big batches of food & freezing can save you money & if it gets you to eat more whole, natural foods then it's still healthier! this works great with the CSA box or public market idea.
this is probably the most important: buy in bulk! buy bulk dry beans & cook them in a slow cooker rather than getting canned. ignore the fancy packaged grains. fill a sack from a bin at the co-op. bring in an empty bottle, have it weighed for the tare, and fill it with honey, vinegar or dish soap & save the cost of packaging, for yourself & the environment.
know which products are cheapest. millet & barley are usually cheaper than brown rice. green or yellow peas or lentils are versatile, inexpensive legumes & cook relatively quickly. peanut butter is cheaper than the other nut butters, but bulk "mystery butter" is cheaper still. sunflower seeds are a cheap, versatile protien source; you can also sprout them, or grind them with water in the blender for a really yummy inexpensive dairy milk substitute. (delicious on oatmeal!) organic kale is cheaper than broccoli, and sometimes collards are even cheaper. blackstrap molasses costs way less than honey or sucanat. you get the idea.
have dh watch the kids some afternoon, so you can take your time and get a cuppa chem-free decaf coffee & spend a while getting to know your co-op. if you go in cold with a cart full of wiggly bored wining kid(s) it'll be harder to concentrate on getting the really good bargains & you may miss something.
learn to make stock, and really stretch out your produce. you can make use of onion skins, squash seeds & carrot peelings, save valuable money & nutrients, & add wonderful flavor to soups, stews, casseroles & lentil loaf.
make everything from scratch, if you can. get a bread machine & buy yeast and whole wheat flour instead of bread. involve rainy-day bored kids in making homemade noodles or crackers. learn to make yoour own seitan, oatmeal burgers & veggie loaf. blend your own nut milk, or if you're really ambitious learn to make soymilk, yogurt & tofu in your kitchen. grow your own sprouts in a jar on the counter. make use of crock-pots, pressure cookers, bread machines and even the microwave to save you time and sanity. packaged goods are really expensive.
this last thing is also very important: if you don't already have it go get yourself a copy of laurel's kitchen. second-hand if necessary. if you can't find it second hand, look for their smaller, cheaper version, laurel's kitchen recipes. it's full of ideas for eating healthy & inexpensively.
there's a few other things i do to save cash: i clean with baking soda & vinegar instead of expensive & dangerous chemicals, i turn off lights, i walk instead of taking the bus (we can't afford a car anyway) i barter with friends & wear second-hand clothes. guess what, though? a lot of things that are best for your health & the environment are also better for your wallet, too! it takes a while to learn the ropes, but i managed to live fairly green on welfare, so you can afford it too!
hope this is helpful!