Question about the punctuation

Before!

After the quotation mark in British English, before the quotation mark in American English. However users of both variants often ignore their own convention.

A reasonably good summary of the differences and shared usages:

http://www.grammar-monster.com/lessons/quotation_%28speech%29_marks_punctuation_in_or_out.htm

American practice would be before the quotation mark. British practice would be after. Additionally, British usage (in my experience) would not employ what Americans call quotation marks. They would use what Americans call a "single quote" and they would call it an inverted comma.

American usage: One day, she found a wallet with a note saying, “Don’t open me.”

British usage: One day, she found a wallet with a note saying, 'Don’t open me'.

OK, so following the American rules, in a sentence such as: My favourite song is “Stairway to Heaven” (where there is a song title at the end), I should also put the full stop inside the quotation mark?

Yes, i would. Please keep in mind that this is a question of style, and that there are not hard and fast rules. The Grammar Police and the Usage Gestapo were both disbanded several years ago.

One could say that a full stop (British name) goes outside the punctuation; a period (US name) goes inside.

Traditionally - American: "double/single quotes", British: "double/single inverted commas", but in my experience just about everybody in the UK has been saying "quotes" or "quote marks" for at least 20 years.