All great responses thus far
Mine in no particular order:
Find and use coupons, both online and paper versions.
Check weekly store circulars for sales and stock up.
Get said weekly circular early in week or check online, then go through and mark or make a list.
*NEVER* grocery shop on an empty stomach.
Stock up on staples when they are on sale; rice, pasta, canned goods (soups, tomatoes, tomato paste, tuna fish, etc). These things last forever when stored properly and are the basis for many inexpensive meals.
Check vintage cookbooks, online, or ask older family members for recipes using cheaper/inexpensive cuts of meat. Many a housewife had to feed a large family when things were tight, and thus learned a thing or two on how to stretch her grocery budget.
Ditto for learning to use some of the tools of our mothers, grandmothers days. Pressure cookers, slow cookers, slow roasting and so forth can make great meals out of cheap cuts of meat.
Then there is offal; something one does not particularly care for but for those that do things like souse, scrapple, and other "ahem" things often are cheap but can make very tasty meals. Again if one is up to that sort of thing.
Cannot be stressed enough to "WATCH For SALES". Learn how to properly freeze meats (maybe invest in a vacuum sealer or something) so you can stock up on things when cheap.
If you don't know how learn to cut up a chicken; whole birds most always are cheaper than parts. At least for the better quality chicken we find anyway. That "yellow" stuff which is largely water may be another matter. Don't know because never buy.
Finally again think like our mothers/grandmothers and learn to love/use leftovers. If you splurge out on a roast for Sunday, remains can be made into other dishes for rest of week. Carcass from roast chicken, and or otherwise bones can be used to make stock.
Also learn what expiration/sell by dates truly mean. For instance contrary to what some may believe things like yoghurt last far longer than date stamped on carton. Have personally consumed yoghurt months after sell by date without any ill effects. Your nose and eyes are excellent judges of when something is "off".
Make the basics ahead of time in bulk and freeze. When there is a sale on for say canned tomatoes and paste, make up a nice big pot of sauce. Keep some aside for immediate use, then freeze the rest in pint or quart containers. One likes to think of tomato sauce as instant dinner in a container. You only need to add some sort of carb (pasta, noodles, potatoes), maybe meat, fish or poultry and bam! You've got dinner.