When stating the title of a book in a essay, italics or apostrophes?

Italics is the usual style for books. The reason is just that it is a convention in scholarship and the book publishing industry; but in journalism, such as the New York Times, they often use quote marks (and roman type) for book titles. Larger works or full-length works are in italics and shorter works (essays, chapter titles, songs, poems) are in roman type and quote marks.

In the old days of the typewriter, writers used to underline the titles of books because it was not possible to make italics on a typewriter. Now, with word processing, we can make italics.

I'd go for italics. The title stands out better, in a page of text, because it's in a style that's different. If you use apostrophes, the typeface is the same as the text, so that doesn't catch the reader's eye.

Either - the only rule is to be consistent. I prefer italics.

Italics. This is standard in scholarly publishing. You can look up things like this in the Chicago Manual of Style.