Does punctuation go inside of quotation marks?
No it does not. See explanation for details.
Quotation, title or whatever you put in quotation marks is a full text and does not need any additional punctuation. If you want to put any punctuation in the sentence, it goes outside the quotation marks.
Have you ever read "Hamlet" ? - here question mark comes after quotation mark to indicate that the whole sentence is a question.
I like "The Lord of the Rings", what about you? - in this sentence the coma comes after the title to sepparate the main sentence from the clause.