The supermarkets here should only be a last resort or where you get staples that are subsidized and the same price everywhere.  That's eggs milk, white flour, and sugar, and to some degree derived products.  The rest of your shopping should be done at the shouk, specialty stores that do both retail and wholesale, and specific "dealers", many of whom have a pickup point in the larger cities or delivery.  Some few supermarkets have low prices on particular items.  They're usually in very low income areas or way out in industrial areas, so have a look in them.

The Shouk (Market)

Almost every town has a shouk.  This is where local growers or enterprising folks rent a space and sell their goods.  Because their overhead is lower, and the turnover of product is fairly quick, their prices are considerably lower than the supermarkets.  Sometimes they are half the price.

For instance, in the city of Haifa, boneless skinless chicken breasts are between 28 and 35 shekels per kilo.  At the shouk, they're typically 4 kg. for 100 shekels or less.  Turkey breasts are even lower, at between 13 and 18 shekels per kilo.  Wings are usually around 8-10 shekels per kilo.

Specialty Stores

These might seem like the last place you'll find a bargain from the outside, but some are suppliers of restaurants and short order places.  You will see large cases of certain items in clear view, but that's the tip of the iceberg.  Even better, once they get to know you, you'll probably get a discount.

What's especially important to you as an African are the ones owned by Christian Arabs.  Christian Arabs are closer in culinary taste to African.  Don't ask me why because I have no idea.  In the "uadi" or in Christian owned stores downtown are the only places you're going to find Spoons brand vegetable ghee (unhydrogenated, mild heat expressed palm kernel oil), home or nearly home preserved okra, and kohlrabi with the greens still attached.

Then there are the smoke houses.  Some places smoke meat and sell it to restaurants, but they have a retail store as well.  These are the best places to get smoked meat.  I wouldn't bother with anyplace else.

Russian stores tend to carry products that other places still don't even though the Russian population here is very large.  There, you'll find the larger bags of baking soda you need for making hominy, cleaning, or cooking with non refined sugars.  They also usually carry Maggi cubes, canned shrimp and octopus, and brands of mayonaise that don't contain any soy.

Dealers

Some restaurants well known for particular things such as meat or seafood, were opened by the suppliers of said items.  So if they have fliers or cards for the supplier, this is why.  Pick one up, and call them for their current prices.  They tend to have a small markup over wholesale, but the prices are going to be much better than even the shouk.  This is also the case with "supermarkets" that are way out in the industrial areas, and aren't really supermarkets.  The big ones are, but the smaller ones named after suppliers and not store networks are actually just the retail face of a wholesaler.  Go in, and see what's on sale.