I HEARD SOMEWHERE A FEW YEARS AGO, THAT IT WAS ACCEPTABLE TO SAY:
"ME AND MY FRIEND WENT SHOPPING".  OR YOU CAN SAY " MY FRIEND AND I
WENT SHOPPING".  ANOTHER EXAMPLE, "AT THE BALLGAME, ME AND JOE BOUGHT
A HOT DOG". OR YOU CAN SAY, "AT THE BALL GAME, JOE AND I BOUGHT A HOT
DOG".  IT IS MY UNDERSTANDING THAT BOTH WAYS OF USING "ME OR I" IS
PROPER.  AM I CORRECT?

Hello don3606,

If you're speaking in casual conversation with someone who doesn't
care about or use standard grammar, then "Me and my friend went
shopping" or "Me and Joe bought a hot dog" may be appropriate. 
However, I don't believe that you'll find a grammarian who would
approve of these sentences in formal writing or speech, such as in a
school assignment or a business presentation.

As the following pages indicate, standard English grammar requires "I"
instead of "Me" as a subject in these sentences.  (While "I and my
friend" and "I and Joe" receives approval from AskOxford, I concur
with your construction of "My friend and I" and "Joe and I" as the
compound subject of these sentences.)  To find the standard grammar,
think of how these sentences would sound without your "friend" or
"Joe."  You wouldn't say, "Me went shopping" or "Me bought a hot dog";
you'd say "I" instead of "Me."  Accordingly, in standard grammar, you
wouldn't say "Me and my friend went shopping" or "Me and Joe bought a
hot dog."

"Which is correct: 'my friend and me' or 'my friend and I'?" (Ask the
Experts: Frequently Asked Questions: Grammar)
AskOxford
http://www.askoxford.com/asktheexperts/faq/aboutgrammar/meandi

"Dr. Language: Are You and I You and Me?" [paragraph beginning with "A
common ungrammatical dialectal construct in U.S. English is "Me and
Maureen ate all our kohlrabi."]
yourDictionary.com
http://www.yourdictionary.com/library/drlang002.html

"1. Grammar: Traditional Rules, Word Order, Agreement, and Case: § 52.
pronouns, personal" (The American Heritage Book of English Usage,
1996) [paragraph on "personal pronouns in compound subjects"]
Bartleby.com
http://www.bartleby.com/64/C001/052.html

- justaskscott


Search strategy --

Searched on Google for:

grammar "me and" "and i"
grammar "me and" "and i" bartleby

Yeah, well some people believe it is okay to write in ALL CAPS.  Some
people are just plain wrong.

Both are not correct if you ask a grammarian or a manners "expert".

In both examples, "me" is being used as a part of the subject, but
"me" is the objective form if "I".  (Objective, not meaning offensive
but that it is the object of the verb:  "He/she kissed me.)  "I" is
the correct form for the subject in both examples, the subjective
form:  "I went shopping."  "I bought a hot dog."

This is where the manners expert comes in.
As a German expression puts it:  Only a donkey names himself first.
It is considered impolite to say  "I and you" or "me and him". 
Both your alternative examples respect this:  "My friend and I ...",
"... Joe and I ...".
That said about correct usage, the incorrect examples ("me and ...")
are nonetheless completely understandable since the word order
(subject, verb object) are clear.
And since we are on the subject of grammar, if you say that you and
Joe bought "a hot dog", then you both only have one for both of you to
share.
In that sentence, I said "you and Joe", putting you first, as the
person being addressed  - spoken to.  This is again a matter of
manners, to mention first the person being spoken to.
AND:  
"... both ways of using "me or I" is proper."  ".. both ways ... are ,,,"
AND:
"... using 'me or I' is proper."     "Me or I" joined between
quotation marks would be one expression.  Writing "me" or "I"  would
be more correct, indicating that each word is an example and not a
part of the sentence.
"Using me or I is ..."  without the quotation marks would lead the
reader to expect that "Using me ..." was a part of the sentence, as in
"He was using me."
So it is correct and important that you used quotation marks, and any
reader would have understood correctly.
English is forgiving in this respect, which may explain why so many
grammatical errors can creep into common usage.  The correct meaning
remains clear.

I hope you found this helpful  (and hope that no one feels the urge now to coorect

I made a comment on a similar question earlier. As 'myoarin' has said,
'me' is not a subjective pronoun and thus cannot serve as the subject
of a sentence.

Can you imagine, for example, saying something like the following:

"Me like ice cream."

This sounds strange, right? So then why would anyone use 'me' as a subject?

Some have suggested that in common parlance it is OK to say "me" even
though "I" is grammatically appropriate.  I will not try and open that
can of worms, as there seem to be many opinions on both sides of the
issue. But, without any doubt, "I" is grammatically correct.

Finally, the context also matters. If, for example, you are writing a
novel where you are trying to capture conversational English, you will
probably want to say "Me and my friend.." rather than "My friend and
I..." because even though the latter form is grammatically correct,
the former is more common to everyday usage and people can more easily
identify with that form.

I did it again, write a comment and then post it much later.
Just as well, Justaskscott's answer is better and definitely more
authoritative than my comment.  I just wanted to explain that I did
not see it first and then use it.

By the way, does an open comment box burden the GA system?

I agree with the others: although some "non-standard" usages can be
defended based on frequency of usage, this isn't one of them. "Me" as
a subject pronoun in the sentences given should be corrected to "I".