Today, my dog and I were talking about grammar, and now I've started to think should that be "my dog and me were talking about grammar".
I also know a man, a former school English Language teacher, who insists that "you and I" is wrong and that "you and me" is right.

So, anyone like to take up the challenge on the usage and thus put 'me and my dog' out of our misery?

It is dog and I were talking if my memory serves me correctly. I seem to remember that one knocks off the first part then see how it sounds, you would not say "me was talking", it would be "I was talking".

Now we wait for Sara who will give us the correct answer.

If I am wrong in agreeing with you, Sue, then Sara can correct both of us. You are absolutely right in what you are saying about dropping the other person, or in this case, dog.

Yes, Sue, I reckon thats right.
"My dog was(sing.)talking. I was (sing.) talking. My dog and I were(pl.) talking."
Well done class!

The dog actually, is a red herring (in fact she's half labrador, half collie and half greyhound).

It is the 'you and I' or 'you and me' that I'm trying to work out.

And which is more proper for Sue to say: 'ren and I' or 'ren and me'?

Ren and I were talking, never Ren and me.

Unless you were being talked to, right?

"Ren and I were talking." Correct

"Sue was talking to Ren and me." Correct
"Sue was talking to Ren and I." Incorrect

I hope I didn't muddle things up, but the above is where I get into discussions with people about which one is correct.

In the case you pointed out, ClaraSue, even if there were ten others listed in the conversation, I was taught to remove all but myself. Correct?

Agreed. I have been corrected on numerous occasions by people who thought that the phrase "____ and I" should always be said that way. I was taught that if you dropped the persons name in front of you, would it make sense?

EX: would you say "Sue was talking to I"? No, therefore, you would say "me" in this case. Does that make sense?

If you can remember that "I" is a subject pronoun that performs the action of the verb, while "me" is an object pronoun that receives the action of the verb, this becomes easier.

Subject pronouns:

I
You
He/She/It
We
You
They


Object Pronouns:

Me
You
Him/Her/It
Us
You
Them