Quotation marks are very useful punctuation marks, but the grammar rules that come with them are complex and numerous.
Let us start with the basics. Quotation marks are used to indicate that a particular piece of text has been taken from another source. They also indicate when a character is speaking — they convey dialogue. "Salutations! How are you doing today?" asked the boy.
Note that when a quote is placed inside another quote, use apostrophes rather than quotation marks for the inner quote. "He said, 'I don't think so,' but it wasn't any use. She didn't believe him."
If inserting something into the quoted material that wasn't there before (or even just to capitalize a letter that was lowercase in the quote), use brackets to indicate that this addition was not part of the original quote. "I think this book is amazing." would turn into "[T]his book is amazing."
If omitting a portion of the quoted material, use an ellipsis to indicate that something has been removed from the quote. "This book, in my opinion, is an insightful commentary on human nature." would become "This book... is an insightful commentary on human nature."