Whether we say " my brother and I" or " my brother and me" depends on what those words do in the sentence.
If these or similar words are the subject of the sentence, the correct pronoun is " I."
Subject:
My brother and I work on the farm.
If the words serve as an object, the correct pronouns is " me."
Object:
Our wicked stepmother hated my brother and me.
Eventually some good luck came to my brother and me.
The confusion arises when the pronoun is paired with a noun, as in " my brother and me."
You can clear up the confusion by mentally taking away the noun:
(My brother and) I worked on a farm.
No one would say, Me worked on a farm.
Our wicked stepmother hated (my brother and)me.
No one would say, Our wicked stepmother hated I.
Eventually some good luck came to (my brother and) me.
No one would say, Eventually some good luck came to I.