If you were asking whether it is more correct to say "John and I" or "John and me", the answer would be simple: it depends on the sentence. "John and I" would be a subject, "John and me" an object. To see which one best fits in any given sentence, simply imagine how it would sound if you omitted "John and":
John and I went to the zoo. (Because we say "I went to the zoo", not "Me went to the zoo".)
Did you see John and me? (Because we say "Did you see me?", not "Did you see I?".)
Using the same logic, you could write "John and we went to the zoo" and "Did you see John and us?" However, those constructions just don't sound very natural, probably because we simply don't need them very often. More common, I think, would be "We went to the zoo with John" (or "John came with us to the zoo") and "Did you see us with John?"