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.

http://www.macmillandictionaries.com/MED-Magazine/January2004/15-feature-ukus-shopping.htm

Hi,

I would like to increase the list of almost "similar" words.

And how about Grocery? What's the difference?