Grains are normally used to make cereal. Cereal is a category of breakfast foods that use grains. These are normally small pieces of processed grain that are consumed at breakfast time. "Wheat" is a grain and when people process that and make small pieces intended to be served with milk, it is called cereal. Cereal can be made of different grains, such as wheat, oats, or rice. Rice cereal: Rice Crispies Oats: Cheerios Wheat cereal: Frosted Mini Wheats

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Grains refer to, as you said wheat, rice ect but could also be used speaking for example of grains of sand or other similar things. While most people associate "cereal"with breakfast foods. Its also a word used in agriculture as I think a general term.

To me (American) cereal is just the breakfast like Frosted Flakes, Raisin Bran etc. Grains are like “whole grains” quinoa, spelt, rice, whole grain bread etc.

They're really similar, but in the US you also hear cereal used most often of prepared breakfast foods, like flakes or puffs. So we'd say "I'm having cereal for breakfast" and most often the person means they're eating a product like Raisin Bran or Cheerios, in a bowl with cold milk. In stores there is usually a "Breakfast Cereal" aisle.