I'm on a pretty strict budget too - I'm a single mom and I bought a house the year before last. We recently had a pretty big tax increase and my already tight budget is even tighter. I notice that many of my friends (childless and renting) seem to not really understand my financial position and often do think that I'm just ignoring them. With about $50 expendable income a week, it doesn't give me much wiggle room! A scholastic book fair or a field trip can sometimes eat into that quickly.
Anyway, for food, I always look at Aldi first - I don't know if you have one in your area. There I can buy almost all of my groceries with the exception of some of my produce list. The Asian markets are so cheap and have great selection. PriceRite is another store near here (owned by ShopRite) where I can pick up some of the more specialty items I purchase (like San Marzano tomatoes for marinara sauce or capers).
I try to buy as little processed food as possible and try to make as little as possible go to waste by cooking, dividing into portions, and freezing. It's the way we extend the shelf life of so many of our meals. We tend to eat a lot of soups, chilis, and stews and use meat sparingly by stretching our food with other vegetables. An example of this is a
Chili which uses 10 oz of meat for about 20 cups of chili or by shredding vegetables into burgers, meatballs, nuggets, burritos, etc.
Cheap staples are beans, rice, potatoes, pasta, frozen vegetables, canned fish, and eggs.
I find that just by freezing, I'm able to really see a reduction in food waste and that keeps my pockets much happier.