Q. When referencing the journal, if the journal title starts with "The", do you put that in or leave it out?

You will need to check with the style you are using, but generally speaking styles which include the journal name in full (like APA, MLA, Chicago and Harvard) will require the entire title including all of the "little" words, while styles that abbreviate the journal name (like AMA and IEEE) will only include the abbreviated forms of the main words.

So if you are using full words for your journal titles, use all of them.