Should I replace "me" and "I" with "he" or "she" when quoting?

If I quote someone saying "me" or "I", when I quote them should I replace them with "he" or "she"?

For example, if someone says:

I went to the movies.

Should I quote that like this:

He/she said "He/she went to the movies".

Or this:

He/she said "I went to the movies".

Your third example is correct. Quotes are composed of two parts; the speaker tag to identify the speaker, and the actual quote itself. For example:

Einstein said "A person who never made a mistake never tried anything new."

Or for your example:

Sally said "I went to the movies."

But you can also paraphrase what they said and integrate it into the sentence without quotes, such as:

Sally said she went to the movies.

You would certainly never use your example of Sally said "Sally went to the movies", because Sally did not in fact say that. When quoting, everything inside the quotes must be what the person said.

If you are supplying a direct quote, you quote the subject exactly.

If you are paraphrasing, then you can change the pronouns to make things more consistent.

An example of a direct quote:

When asked what he did last night, Dave said "I was at the movies."

Paraphrased, it would look like this:

When I asked Dave what he did last night, he said he was at the movies.