Is it John and me, or, John and I?

John and I when they are the subject of the sentence;  John and me when they are the object. 

It is John & I, only if I is followed by a verb.

Don't know what the hell everyone else is going on about, but it's definitely 'John and I'. Getting into the habit of saying 'me and John' is a bad habit and also atrocious grammar. 

It is John and I if they are the actors, for example, John and I played basketball.

It is John and me if they are the recipients. For example, the pickup basketball team gave John and me a shellacking.

Rule of thumb is to remove the reference to the other person, and consider how you'd say it if you were just referring to yourself:

Example: if you take out John's name, would you say "I am going to do something" or "Me is going to do something"? 

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Depends. 

In some cases it's ok to say me.

John and I walk around the lake.  Subject with verb.

Jill gave John and me a sandwich.  Predicate no verb. 

Depends on the context.Way I learned it was: take the other person out of the sentence. Would you use "me" or "I"?  "She gave me a sandwich." So it's "She gave John and me a sandwich".