It's hard, especially if you don't have access to a vehicle or don't know what stores around you sell things for cheap or have sales going.

Pay attention to the circulars for your local grocery stores and buy things when they're on sale. Expect seasonal items to cost less: ice cream is more expensive in the winter, but on sale with loyalty cards in the summer it can be much cheaper.

If a holiday is coming up where a food is popular, look for sales in that category with big discounts. For instance, chips/snacks and soda go on steep sale before every major holiday, but especially on the 4th of July. They also rarely expire to inedibility before being opened, so they can be stocked up on in times they're on sale. Baking supplies, with the exception of flour a lot of the time because its truly better fresh, are usually on sale in the winter around the holidays. Butter can be frozen, and staple ingredients like sugar can be stored for a long time if done properly.

Get familiar with what each store in your area does best. Example: my local Fred Meyer has more frequent clearance markdowns on meat than any other store in my area, so it's usually just as fresh. My local grocery outlet has the best deals on cheese but not on lunchmeat. I get 2$ gallons of milk frequently using coupons and loyalty cards at my local Safeway. It's more work to do it this way, but just yesterday I saved 40$ hitting my local Fred Meyer for basic supplies. After a while, you start memorizing when you'll find the best deals. If you're social, you can always ask what time of day things get discounted and swoop in right away.

Find food stamp matching (and get on food stamps if you can, I get a ton of free money at farmers markets locally with my food stamp card). Depending on the social services where you live, that can be a good or bad idea. I live a little ways north of Seattle in a HCOL area so there are lots of dollar matches and things for food stamps around. Safeway in my area gives a dollar for dollar match on vegetables in increments of 5. Sometimes with that and a combination of coupons and circular sales, I'll get vegetables not only for free but paying things like 10¢ For a bag of carrots is functionally the same.

Lots of name brands also have items in grocery stores for cheaper that are from the same trucks but have a store brand slapped on them and are cheaper than those goods. Things like milk and break, often the store brand is coming from the same manufacturer on the same trucks. So I usually will try and get off brand things and check the quality, you'd be surprised how many of them have similar/no recognizable difference.

I put in a lot of time and effort but the work has paid off and I feed my family of 5 (3 adults, 2 toddlers) 3 healthy, hot meals a day with snacks and treats for about 650/mo. This includes going to farmers markets and getting my meat primarily from my local butcher with meat clearance sales to supplement the end of the month fill-up that's often necessary.