Do titles of short stories and essays get put in quotes or underlined?

It depends on how they are published. If they are published in a compilation (many essays in one book, or many short stories in one book), then you put the title of the short story or essay in quotation marks, followed by the italicized name of the publication where it is compiled.

If the book or essay is published by itself (some are published this way if they are popular enough) then it is the only title, and you would italicize it.