The easiest way is this:
We (or someone and I)do . Example: my father and I, Dan and I, the group and I, etc.
They (or nouns in the plural / names / etc.) do. Example: the boys, Mary and Ann, those schools, etc.
He (or: a name - masculine gender, a masculine noun, etc.) does Example: Paul, that man, etc.
She (or a name - feminine gender, a feminine noun, etc.) does Example: Mary, her sister, etc.
It (or whatever isn't a person) does Example: That dog, a tree, hope, New York, etc.
There's more to say about the use of subject pronouns (I, you, etc.) but you asked only about the difference between do and does, so this will do. :D