A grocery store (AmE) sells a wide range of foods. In the UK, the term "grocer's shop" is a little outdated - it used to refer to small, local shops which sold a bit of everything, both food and non-food but only a very small range. In BrE, a "greengrocer's" sells fruit and vegetables but some are branching out into chilled items like dairy, cheese, yoghurt etc and also packaged goods like breakfast cereals and biscuits. There are still some which sell just fruit and vegetables.
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