The real secret to cheap groceries is knowledge - know your stores, know your prices, know the sales, know who's good about marking stuff down while it's still usable. You can save money and get better quality - more bang for your buck - if you scout out the local stores and watch the ads.
The biggest markdowns are usually in the meat department, just because meat tends to be more expensive than other ingredients. Some discretion is necessary: you don't want to buy ground meat that's been marked down, for example. But things like steaks are already aged anyway - what difference does one or two more days make?
Careful examination is also needed - you don't want anything that's discolored or smells off. A good store will mark it down before it suffers from either of these symptoms. Don't take any chances - if it doesn't look good or smells bad, pass it by. If you see a lot of low-quality packages in the marked-down section, maybe buying marked-down stuff from that store isn't such a good idea.
My favorite places for marked-down meats are the Westwood Apple Market at 47th and Mission Road, and the Price Chopper on Roe in Roeland Park, just south of I-35. Both are good about discounting stuff before it goes bad, which is what you want. As a rule, stores located in more affluent areas tend to be better about mark-downs. (I don't advise buying marked-down items - unless they're sealed and dated brand names - from inner-city markets)
Don't forget to check the store's regular mark-down section for canned or packaged stuff at up to half off the regular shelf price. A dented can is no problem as long as it's not bulging and doesn't hiss when you open it. Most boxed stuff has inner packaging these days, so crushed boxes aren't a big deal as long as the contents aren't exposed to the air. Like the marked-down meat, use common sense and you'll be fine.
Discounts on bakery items are easy: there are several Oroweat Bakery Outlets in town providing quality products at discount prices. I prefer the location on Shawnee Drive west of I-635, but I'd imagine other locations have much the same deals. Other bakeries have thrift stores as well, but I prefer the Oroweat products.
Another great place for discounts is Dirty Don's in Raytown. You never know what you'll find there - I've gotten everything from cookies to rolling papers to guitar picks - but it's all dirt-cheap. (since it's mostly salvage, it can be a little dirty as well, hence the name)
Please feel free to share your favorite KC-area discount stores in the comments.