In English language, no two words are exactly the same. True, they can have similar basic definitions. But how they are used is different.
For example, 'shop' can be frequently used as a verb as in 'I like to shop at the mall.' But when 'store' is used as a verb, it means to keep something at a safe place. Also, 'store' most often is used to mean the 'grocery store' but in the U.S. they don't use 'shop' in this sense very often. Unless you say something like 'the corner shop' implying a smaller-sized store which is conveniently located in a neighborhood that people would walk to.
My point here is, words like 'store' and 'shop' have many definitions & usages. My suggestion is to go to an online dictionary like Cambridge and look at different examples to help yourself understand different meanings.
Generally, "store" would be the normal American usage, and "shop/shops" would be the normal British usage when you are referring to a grocery shop or supermarket where you buy food and other things (such as Tesco, Safeway, Wal-Mart, etc). However, in Britain most people use the plural "to the shops" or "down the shops", whereas Americans tend to use the singular "at the store".
Of course, if I was going to the Tesco in the UK, I would most likely just say I was going to the supermarket.
Hope this answers your question.