I recommend Big Lots. They have both pet food and people food. You have to be willing to take a chance, because there are lots of items that are uncommon brands or flavors. However, the price difference is worth getting the occasional lemon. The bad thing is that most items are only available until they run out. Our Big Lots also has discount bread. It is fresh, name brand, and less than half the price of the supermarkets. I caution you, though, it is easy to get carried away by the prices. You get a lot for your money, but sometimes it is tempting to buy stuff you don't need because it is so cheap.
I also buy discounted meat. Most grocery stores put meat on sale when it is near its expiration date. Like today, I bought two pounds of fresh ground turkey breast at Kroger for $1.25 per pound. I packaged it into smaller servings and froze it. I got a discounted five and a half pound package of extra lean ground beef at Super Wal-Mart for about $12.00 a few weeks ago. I've been purchasing meat this way for years, and no harm has ever come to me. Just make sure you cook or freeze the meat immediately.
I also shop at Aldi's, and sometimes Save-A-Lot. Aldi's is cheaper, Save-A-Lot has more variety. The fresh fruit and vegetables at Aldi's are so cheap you will not believe it until you see it for yourself.
I grew up eating the brands from Aldi's, so it tastes O.K. to me. Not everyone likes the store brand, so you may want to stick to things like produce, milk, eggs, etc.
I plan on buying a small deep freeze soon. I want to learn how to prepare fresh fruit and vegetables for home freezing. I have access to a farmer's market, and I really want to take advantage of the low prices and good quality.