Ah, the quotation mark. It takes many forms…in the air, overused, underused, absent, and just plain wrong. Some use it for “emphasis.” (Hint: don’t.) Some put quotation marks common phrases in signs or marketing collateral. The short answer on all this is, of course, that quotation marks are marks for notating quotations. Not for emphasis. We might use them to set off a phrase someone said or their misuse of a phrase (such as a liar’s version of the “truth,” for example). But otherwise, they only work for quotes.
But within that, how are they treated? In American English, double quotations go around an actual quote. Punctuation (periods, commas, dashes, etc) go inside. It’s different in British English, but that’s beyond the point. When in doubt, consult this APA table.
There is an argument for logical punctuation, but as an old English major having two major publication styles chiseled into my head, this sort of thing makes my skin crawl.