Are beans considered vegetables?  I understand they are classified as
a legume.  What is the difference between a bean and grains?  Both are
cultivated plants and sometimes beans are considered vegetables but
grains never are.

Beans are vegetables.  They grow on plants that are legumes.  Grains
are the seed of grass plants (as a rule, there may be some
exceptions).

If you search with   define:legume    define:vegetable    define:grain

you will get definitions of each word.  Some for "grain" will, of
course, refer the sand and other things.