If you end a sentence in a quote where does the period go?

There is a difference between US and British/Canadian punctuation styles.

In the US

Trailing periods and commas always appear inside the quotation marks

ex. "Let's go to the zoo."

ex. The spelling of the Word is "ampersand." - He said, "Go now," and turned away.

But question marks follow the same logic logic as below.

In the UK and Canada

The punctuation follows the logical extension of the quote.

The period or comma goes outside the quotation mark, except where the period is part of a quote.

ex. He said "The day is long." - The movie was called "Benji".

In either version of English

If you are quoting a question then the ? will go within the quotation marks.

ex. Sally asked, "Where are you going?"

If you're asking a question about a quote, then the ? will go after the quotation marks.

ex. Did Sally say, "We are going to the zoo"?

Note: There is never any double punctuation used.