Is the correct grammar my mother and me or my mother and I?

John and I is correct only when the sentence is using this phrase as its subject, i.e. it is used in the subjective case. Example: John and I prefer Hot Dogs. When the sentence they are in uses them as an object of a verb, only John and me is correct. An example of this is, "The boss fired John and me today." An object of a preposition like "to" or "for" or "from" also uses the "me" form, as in, "Do this as a favor for John and me." This may sound confusing and technical, so I'll make it very simple and foolproof. Whenever you are about to use either phrase, try saying your statement with only the "I" or "me". Leave out the Words, "John and". You have just raised a much easier question: "Which is correct, I or me?" If you have a knowledge of the English language at the level of a 5 year old or more, you will immediately know the correct answer.
For example can you handle these?
1. I

prefer hot dogs. / Me

prefer hot dogs. -- correct usage is "John and I

".
2. The boss fired I

today. / The boss fired me

today. -- correct usage is "John and me

".
3. Do this as a favor for I.

/ Do this as a favor for me

. -- correct usage is "John and me

".
Finally, if the meaning of what you are saying would turn to nonsense when applied to only one person (the I or me) you may have to try "we / us" instead of "I / me".
The thing you have to remember now is that WE goes with I, and US goes with ME.
Trying it in the sentences above:
1. We

prefer hot dogs. / Us

prefer hot dogs. -- correct usage is "John and I

prefer hot dogs".
2. The boss fired we

today. / The boss fired us

today. -- correct usage is "The boss fired John and me

today".
3. Do this as a favor for we.

/ Do this as a favor for us

. -- correct usage is "Do this as a favor for John and me

".
Read this answer over several times if necessary. Once you catch on, making the correct choice will become very automatic.

Practice by thinking up puzzlers to test your friends, and then tell them the reason for the correct answer. Here are a few to get you started:
1. The award should have gone to John and ___! (Hint: to I

/ to me

)
2. Your decision really made John and ___ very angry. (Hint: made we

/ us

angry)
3. What if John and ____ treated you that badly? (Hint: What if I

/ me

treated you ...)