Object pronoun: me and John, or John and me?

Can't go wrong with '___ and me'.

As Peter Shor and John Lawler have mentioned in the comments, grammatically either order is correct. However, English-speaking children are taught to put the pronoun referring to themselves last (the reason given being that it is more polite to put others before yourself), so I would agree with skaaptjop that "__ and me" is the safest way of saying this (this of course only applies to the object position; for the subject "__ and I" would be used instead).

Despite this rule of etiquette, it's not surprising to hear people using phrases like "me and you" or "me and my friend".

I remember being taught in grade school (in the 70's) that "me and ________" was the correct form for the object of a sentence, but I have seen both used since. Also, it was grade school so maybe they said both ways were acceptable and I have forgotten in the interim.

When writing formal emails to clients, and I am asking them to contact us if they have any issues, I use either "me or John" OR "John or me".

How I personally differentiate between the two is by putting the preferred contact's name first. If I'd prefer the client to contact John, I would put his name first. As both are grammatically correct (per the above responses) I have no preference other than that!

As for "John AND me"... I would probably write it this way just for the formal looking nature of it (damn you "John and I"!) - but I wouldn't lose any sleep over it either way.

Unusually enough it would seem, I was actually taught that when using 'I', I comes second, but when using 'me', me comes first, so it would be 'Bill and I' for the subject of the sentence or 'me and Bill' for the object of the sentence. I don't recall ever being taught that it was more polite to place yourself last in the listing.

I was also taught from grade school through college that "I" always comes second, but "me" always comes first. In a nutshell "XXXXXXX and me" is never correct. I majored in English and have worked as a salaried writer for almost 30 years, and I can't believe how many resources there are online espousing use of "and me." (Have I discovered "Fake Grammar"?)

If you think about what any "me and" statement is communicating, it makes sense to state "me" first. For example:

"I hope you'll join me and the leadership team at our annual meeting."

Stating "me" first makes the statement sound sincere. You'd never say, "I hope you'll join the team in going to that thing I won't be going to." Why would you hope someone would do something with someone else that you're not involved in? You wouldn't, so saying you would have some hope about it is insincere. You're only going to hope someone attends something that you will also be attending, ostensibly because you hope to see that person at the event. The other people being there are beside the point. You can only express your own "hope," which is that "you'll join me."