John and I are going to London tomorrow.
Can you help me and John?
Would you like to come to the party with me and John?
(You could also say "John and me".)
I am going to London.
Can you help me?
Would you like to come to the party with me?
If you take "John" out of the sentence then I think it's much more obvious whether to use "I" or "me". So if you are trying to decide whether to use I or me, just try to make the sentence without the other person and see what sounds right.
However, you will hear a lot of English people say "me and John" instead of "John and I". Some people would say that it is acceptable nowadays but others would disagree! It is a little more acceptable in informal speech than in writing. Personally, I think it's better to say "John and I are going to London." When I was a child, my mother corrected me if I said "me and John". She even explained the rule to me! I was probably only 6 years old and I still remember that conversation!
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