Yes, it should be ".... and me" if you are saying, "This is a picture of John and me."
It is correct to say "John and I went to the fair." It all depends on if you are the subject or the object.
You can always test this by taking "John" out of the sentence.
ie. "This is a picture of me." (You wouldn't use "I " there.)
"I went to the fair." (You wouldn't say "Me went to the fair."