First, all the basics everyone already knows: don't shop hungry, avoid bringing along the kids, pay attention, avoid impulse buys.
Always shop with the sales paper. When I get my sales paper, I highlight any items that are a good price, and that we need. If apples are on sale for $.54 a pound, I plan on picking up some for snacks, baking, applesauce or even to freeze or dehydrate to use later. If pasta is on sale for $.75 a box, I circle that and plan on buying a lot. If you don't receive the sales paper in the paper or mail, grab it when you get to the grocery or check online.
Make a list of items you definitely need and try to stick to it as much as reasonable possible. You don't want to just waltz in and throw whatever looks good at the moment or you'll end up with lots of snacks and sweets, and spend a lot more than you budgeted. On the other hand, you don't want to be so rigid that you pass up a fantastic deal on an item you regularly use just because it wasn't on your list. Some people also find great value in making a meal plan around the best sales in the paper, and this is especially good for those who have very busy schedules so you can avoid the after-work "what's for dinner?!?!" panic.
Shop first at the cheapest grocery store you can, then work your way to the more pricey ones to get anything that you can't find at cheaper venues. About every three months I shop at a really good salvage grocery and stock up on whatever they have at a good price, usually breakfast cereal, salad dressing, pasta, and juice. I save weekly for this grocery trip and plan to stock up on three months worth of food to get me through to the next trip. It's an hour away, so if I lived closer I'd shop more frequently. I often shop at Aldi or similar discount, no-frills grocery store. That's the best way to buy dairy, eggs, baking supplies and canned soups and veggies. Sometimes I shop at Kroger, Meijer or similar big grocery store chain. I get everything left on my list after the cheaper stores.
Sometimes bulk purchases save lots of money. Sometimes they don't. Sometimes generic brands are a fantastic deal, other times they're the same price as name-brand. So always, always, bring a calculator (this is the one benefit to everyone always having their cell phone on them). Divide the price for the item by the units of the package (pound, number, ounce, etc). Check this for the big package and the small package, the generic and the name brand. In this way you'll know which item is actually the cheapest, and can purchase accordingly.
Generic is often just as good as name brand, but not always. Especially for things such as salad dressing, the recipe may be better with one brand than another, or there may be a limit to selection in the generic (then again, since when do we need 30 different varieties of ranch dressing?). However, do try the generic. If you like it, you'll likely save a lot of money over the long haul. If you don't, you're probably not out much money. If you have a bad experience with one store or knock-off brand, try another. Items such as white sugar are the same regardless of brand. Buy the cheapest per pound. Regular all purpose flour is pretty much the same. Differences in flour come with buying bread, cake or other specialty flours, and are really not a big concern for the casual home baker. I'll bet your kids won't hate you because your cake was just moderately fluffy instead of being extra fluffy.
Before you shop anywhere else in your grocery store, check the clearance bin. If the item is nearing it's sell-by date, has damaged packaging, has old-design packaging, or is a discontinued variety, it is marked down, often 25-75%. Some stores have better deals than others, but it's always worth checking. See if you have any coupons for any of the items. If so, subtract the coupon value from the clearance price, divide by unit and compare to regular priced items (sometimes the clearance mark down isn't that great and won't be better than generic, even with the coupon).
Bread that is labelled "whole grain" or "whole wheat" is almost certainly not. It just has some whole grain or wheat added, and legally that's a-okay. The only way you can be sure it is completely whole wheat or grain is if it says "100% whole wheat" or "100% whole grain". "Wheat" bread is just a lot of white flour with a bit of bran thrown in and maybe some dye. Don't pay extra for the same basic product.
Store-brands of pasta are usually satisfactory and a great bargain compared to national brands. Many stores even offer whole grain pasta that is only nominally higher than their white pasta. Whole grain pasta is more healthful and filling that white pasta, so it is a great deal. Those pastas that have veggies mixed in are usually a lot more expensive, and I doubt that they are that much more healthful. Buy them if you find a great deal, but don't expect them to change your health.
Pasta sauces vary greatly brand to brand, but some of that can be overcome in cooking. Add seasoning to jazz up a bland sauce. A thin sauce can be cooked down to evaporate some of the extra water yielding a thicker sauce. Do experiment with all of the brands in your price range. There are three different brands that I like, and I buy whatever is the cheapest of those three.
If you buy canned soup, buy concentrate when possible as this is usually a better deal. If you dilute the concentrate with a can of water, halve the price per ounce when comparing the price to full-strength soup. Discount grocery stores usually have satisfactory canned soups in the basic varieties (chicken noodle, bean, tomato, chicken rice and beef vegetable).
Avoid baking mixes. They really don't save that much time and usually cost a heck of a lot more. Also, they lock you into baking whatever mixes you have in the cupboard instead of getting creative with what you have.
Ultra-pasteurized homogenized milk is ultra-pasteurized homogenized milk. Don't pay extra for one brand over another. However, raw milk, organic milk, goat's milk, lactose-free milk, or non-homogenized milk may very well be worth paying more for and is a higher quality product.
Be wary of labels on eggs touting "cage-free" "free-range" or "vegetarian fed". These are tricky labels. "Cage-free" may mean that a thousand chickens are crammed in a building with no access to the outside, but no bars. "Free-range" may mean that the hens are allowed access to a little dirt-patch, but never do because their first few weeks of life they were cooped up inside "for safety" and are now afraid to go out. "Vegetarian-fed" is pretty much a guarantee that those chickens never saw the outdoors. Chickens are omnivores and adore insects and mice. If they see a bug, they are eating it. If they are guaranteed vegetarian-fed, it means they weren't allowed outside and were fed a diet of grain. If you're going to pay extra for eggs, buy from a neighbor or local farmer you know.
Check the clearance meat section for marked down meat. When the package is a few days from its sell-by date, it is often marked down 25-50%. Buy what you can use within a day or two, or freeze for using later. Sometimes the deals are fabulous. Once I found ground hormone-free, pastured beefalo for $2/lb. I bought all they had and froze to use all summer long. When turkeys go on those wonderful deals around Thanksgiving (or hams around Christmas or Easter, or corned beef around St. Patty's), buy one or two extra. I love to pull a turkey out of the freezer in February and roast it, then package it into meal-sized portions (chunks for stir-fry and casserole, shreds for omelets, sandwiches or soups, and cook the carcass down for stock).
Most importantly, the grocery store is not the only place to buy groceries. It's also not usually the best, nor is it the cheapest.
Bakeries can be expensive, but they offer a very high quality product. Sometimes you can find a bakery that is outrageously cheap. One of my favorite bakeries in Chicago sells sub rolls for $.33 apiece and mini cookies 5 for $1. Ask your baker how long they ferment their bread before baking. The longer ferment times means that the bread is better for those with a gluten sensitivity or blood-sugar issues. At many bakeries, you can also find individual rolls (which are often great for sandwiches) or small loaves of bread so you can avoid waste if you're buying for one or two. (If you do have stale bread though, check out this article for ways to use it up)
Farmers markets can be very expensive if you just go in and buy whatever looks good (hint: it all does). If you eat very seasonally, it can be the best deal around. Once at the height of the tomato harvest, I saw a sign advertisting canning tomatoes 50 pounds for $25. Someone asked the farmer what made them canning tomatoes. He said, "Everyone here has tomatoes for sale. This is the only way I can move a lot of them." They were perfect tomatoes at a perfect price. My favorite farmer sells peppers at these prices: small cherry bomb hot peppers 5 for $1, jalapenos 5 for $1, hungarian wax peppers 4 for $1, green bell peppers $.5, red/purple/orange/yellow/black bell peppers for $.75. Another sells small hot peppers $.10/each and small sweet peppers $.15/each. Compare this to the grocery store that sold green bell peppers for $1 and red or yellow bell peppers for $1.50. When zucchini are in season, I often find them 4 for $1. When you find prices like these, stock up and preserve for later (shred and freeze zucchini for baking, dehydrate tomatoes or can sauce, freeze or dehydrate peppers, make pickles out of cucumbers, and puree cooked pumpkin for pies and muffins all winter long).
If you live in farm country, farm stands are great. You get great bargains and the items are very fresh. You also get to peek at the farm and talk with the farmer, two vital steps for responsible eating. You get to support a small farmer and thus save the world from Monsanto, and you keep the money in your community. So many wins.
If a neighbor sells produce in a little stand in their front yard, go nuts! These hold the best deals I've found on produce anywhere, except for growing it myself. I often find tomatoes, peppers, eggplant and winter squash for $.50-.75 each and cucumbers and zucchini for $.25. Again, you're keeping the money in your community and supporting cottage industry.
If eating pasture-raised meats, organic produce or raw milk is important to me, going through small farmers or homesteaders is the best way to go. For raw milk, it's the only way you can get your hands on it. For eggs, I pay $1.50-2.50 a dozen when buying from a small "hobby" farm, versus paying $7-8 through other venues for a similar product. Buying a share of a cow, pig, goat or sheep is often the best way to be able to get pasture-raised/hormone-free/organic meat at a price that's realistic for working and middle class families. You can even go in with friends or family members for a more manageable price and quantity.
Ethic grocery stores are almost always cheaper than grocery stores for ethnic foods. The price differences are astounding. At the grocery store, an 11 oz bottle of sweet roll sauce is $2.79 or more. At my Asian market, a 32.5 oz bottle is $3.25. White rice is over $2 for a pound bag at the grocery, but I just bought jasmine rice at the market for $12 for a 10 pound bag (50 pound bags are an even better deal, so next time I'm going to split a bag with family members). You can also find veggie varieties that the grocery doesn't offer for the same price or less than the same-ol-same-ol varieties at the grocery. Latino grocers are great for finding the less common parts of the animal. My favorite sells chicken feet for $.20/pound so I can make killer stock.
Talk with your local butcher about buying meat from the animals that people dropped off, but never picked up. The butcher often can't sell the meat, so you just pay the processing fee. This is common during deer hunting season. I get a whole deer for around $80.
If you have a bit of space, whether in the backyard, the patio or a windowsill, grow some of your own food. Seeds cost a penny or less. If you limit your inputs of store-bought chemicals, and instead compost your kitchen and yard wastes to make your own fertilizer, you can get practically free food. A penny for a seed, plus 10 cents worth of potting soil, put into a repurposed container such as a coffee cup, turns into a tomato plant that provides 4-10 pounds of tomatoes for your family. Lettuce is really easy to grow, and right now I'm even growing several pots on a sunny windowsill to add to salads in winter.
As you shop, work towards building a stockpile of food at the lowest price possible. Buy non-perishables in large quantity when the price is low. Pull the old ones to the front and put the new ones in back so nothing goes bad. Freeze, can or dehydrate produce when it's in season to use when it is out of season so has to be flown in. Freeze meat in meal-sized portions whenever you find a great price. Eventually you'll be buying not for the week's meals, but to replenish your pantry. At that point, you never have to pay full-price, you can just eat from the pantry while you wait for the next great sale.