When it appears that two punctuation marks are necessary at the end of a quote, many writers scratch their heads.

Consider this example: Can you imagine how people felt when she yelled, “I refuse to listen to this!”

Many people are inclined to put a question mark after the sentence, but we rarely double punctuate at the same location. Instead, we choose the stronger mark, based on a hierarchy. Exclamation points take precedence, question marks are second in importance, followed by commas and periods. So at the end of the sentence above, the question mark yields to the exclamation point.

Other examples of correct punctuation:

Was that an appropriate time for her to ask, “Why are we doing this?”
(When the entire sentence is a question and the quote is also a question, use one question mark, and place it inside the quotation marks. The inside takes precedence over the outside.  Use only one.)

Was that an appropriate time for her to say, “We shouldn’t be doing this”? (The quotation is a statement, but the period is dropped; it yields to the question mark.)

As for the colon and semicolon, remember the standard guideline: Commas and periods always go inside quotation marks, and colons and semicolons always go outside.