Do commas and full stops go inside or outside quotation marks?

It depends which side of the Atlantic you are writing for, and this is one of the clearest differences between American and British style.

American style puts commas and full stops inside the closing quotation mark, always, regardless of whether they belong to the quoted material.

She called it "a disaster," and left.

British style places them according to logic: inside if they are part of the quotation, outside if they are not.

She called it "a disaster", and left. (the comma is the sentence's, not the quote's) She said, "This is a disaster." (the full stop belongs to the quoted sentence)

The British approach is sometimes called logical punctuation for that reason. The American convention originated with typesetting: a full stop or comma after a raised quotation mark was fragile in metal type and prone to damage, so printers moved it inside.

Colons and semicolons go outside the quotation marks in both styles.

Question marks and exclamation marks follow the logical rule in both styles. Inside if the question is part of the quotation — she asked, "Where is it?" — and outside if the whole sentence is the question — did she really say "disaster"?

On which marks to use: American style prefers double quotes with single inside them; British style is mixed, with single quotes common in books and double in journalism. Nest the opposite type inside.

Block quotations — long passages set as their own indented paragraph — take no quotation marks at all.

Pick a style and hold to it. Mixing the two within a document is the visible error.