A supermarket is a large grocery store (US)/shop (Br) where you might find what you are looking for.

Whatever it takes to cook a good meal, you will find it at the grocery store/shop, supermarket, food mart, or at one of the big food chains usually represented all over your country.

English Grocery

A grocery store in Tucson 

Green Grocer's is primarily a place for vegetables, fresh field produce such as swedes (Yes, yes…), tomatoes, potatoes, lettuce, onions, cauliflower, parseley,  horse radish etc.

The Butcher's, Meat Market, Meat Mart or  Butcher Shop, is where you find  minced meat (Br) or ground beef, sausages, sliced ham, leg of lamb, game and poultry.  

7-11

Dairy products are most commonly found in your ordinary grocery store, supermarket or deli (delikatessen), together with places like  7-Eleven  
(7 - 11)  stores or other similar round-the-clock stores/ shops.

SIA-glass

Dairy Shops carrying dairy products only aren't  that common anymore.

Mostly we buy our dairy products at the grocery store,at the grocer's.

Dairy products come from dairies, dairy farms or other suppliers, distributors or wholesalers.

Delikatessens, or delis, focus on/specialize in/ fast food, sandwiches, sandwich items, snacks, beverages, liquor, beer, wine, candy, cigarettes and a limited supply of groceries and chemicals.

The closest thing to this would be a "servicebutik"  in Sweden. So far we haven't seen any wine and liquor though.  

Hemglass
 

The American drugstore is one of the most frequently mistreated, and abused words there are, when it comes to translation from TV, radio and newspapers together with, unfortunately, school text books at all levels throughout Scandinavia.

Drugstores do carry drugs, medicine, pills, hygiene articles, contraceptives and many, many other things, yes.

However, Swedish "apotek" is far from a drugstore. It might resemble the prescription desk at the drugstore. Swedish "apotek" is similar to Pharmacy and Chemist's, and in some instances only, to a drugstore.

In a regular drugstore you find all kinds of goods/ articles/ items, such as cleaning liquids, bathroom supplies, cosmetics, beverages, tobacco products, batteries, cassette tapes, underwear, watches, candy, hair spray and deodorants, snacks, films, and sometimes even cameras, ice-cream and ice. 

Triumfglass

Occasionally, especially in smaller towns all over the U.S., you will find a counter at which you can have a cup of coffee, an ice cream, a sundae, milkshake and a cup of hot chocolate with whipped cream on top. In the drugstore business, as in many other trades, you can see many chain stores such as Longs Drugs, Rexall Drugs etc. In smaller towns the drugstore is usually the meeting place of the local town population. Some people say for sure, that this is where rumor starts, and this is where it ends...

GB-glass