Modern "grocery stores" are actually what you would call "super markets" where you can find a wide variety of durable goods, preserved foods, as well as fresh produce, meat, baked goods, etc. This is a historically fairly modern invention, dating back to around the mid 20th century.

Historically, there would be a variety of different vendors for various foods that one could find in a market, similar to today's "farmer's markets" in the US. Back then there were storefronts that sold foods but primarily they would have been long-lived dry goods like spices, tea, sugar. Things with high shelf-lives and also high prices relative to their size. Around the mid 19th century or so that started opening up as more ways of preserving foods started becoming common and as industrial food production started ramping up (with things like canned foods, prepared mixes, and so forth).

However, to answer your question, people would have gone to a butcher for meat in the early 1900s, or bought (or killed themselves) whole animals. Things like chickens, turkeys, geese, fish, etc. might be expected to be bought whole then cleaned and dressed at home by whomever was doing the cooking for the household. Things like cows or pigs one would typically have gone to a butcher to get cuts. And what was available would be highly variable depending on where one lived and what the regional traditions were.

Someone living in the French countryside would probably find a much different variety of cuts and products available at their local butcher than someone living in, say, New York City, or Chicago, or San Francisco, or St. Louis.

Also I should re-emphasize that butchering techniques and cuts depend on local culture and regional popularity. Different regions and cultures practice different butchering techniques (such as seam butchery versus, I guess you'd call it "bandsaw butchery") and have different favored cuts and preparations. For example, bacon in the US and Canada has long been different from bacon as it's known elsewhere (which tends to be either some type of back bacon, fatback, or peameal bacon). Different meat primals might be cut up in different ways and made up into different steaks/cuts or processed into different foods (sausage, hams, ribs, etc.) depending on local demand, traditions, etc.