I would have given an explanation similar to that of Americo S., but I have noticed that, in many countries, the US for example, 'shop' and 'store' are interchangeable.
We distinguish between the different sizes of stores by referring to 'supermarket' as a larger 'chain' store that usually sells food, and a 'department store' as a very large store, usually in a larger town or city, that sells a wide variety of products.
store = a place where they sell stuff
shop = a small store where they sell stuff ( often owned by a people instead of a company)
market = normally a place where there are a lot of small shops in the same area.
There is a vocabulary difference between British and American English.
"I am going to the store" = American English
"I am going to the shop" = British English
A "market" is usually outdoors and people sell things on tables or stalls.
Note: in British English, we do use the word "store" as a verb. For example, we could say "where are we going to store all this food that we have bought?" It means "to keep / to put".
This link may help you.
I would like to increase the list of almost "similar" words.
And how about Grocery? What's the difference?