Is it correct to use "punctuation outside of the quotations", or "inside?"

Or is it region specific? I was always taught that when ending a quotation, that punctuation remains inside of the quote.

I think he said, "we should go to the store."
Are you sure he said, "we should go to the store?"

As opposed to:

I think he said, "we should go to the store". Are you sure he said, "we should go to the store"?

This is just an arbitrary example off of the top of my head, and it's hard for me to come up with an example for the second usage because it looks completely wrong to me.

I actually got into a small argument with my girlfriend earlier this year because she uses something similar to the second example, and said that's how she was taught (which is why I ask if it's region-specific.)

Anyway, is there a correct usage for ending a quotation with punctuation?

Answer

In American English, commas and periods go inside the quotation marks. Semi-colons, question marks and exclamation marks go inside the quotation marks only if they're part of the quotation. E.g.,

"What time is it?" he asked.

Did he really say, "I don't care"?

So your example should be the following:

I think he said, "we should go to the store." Are you sure he said, "we should go to the store"?