Well, "shop" and "store" are mostly the same.
Therefore, "the store" is a store that is well known both by the speaker and the addressee.
99% of the time that means the grocerynot far from where the speaker lives.
Synonyms for the store
On this page, you will find 30 slang terms related to the store. Some of the top words include: asda, pizzlewick, cat rodeo, Hot Topic, Pierpont, and 25 more.