Apostrophes are used only for contractions and possessives. People often slap an apostrophe on plurals, but this is not correct. Pronominal possessives--"its," "his," "hers," "theirs" and "yours"--never need an apostrophe. The only time "its" will have one is when it is a contraction for "it is" or "it has."
Further, proper noun possessives with an "s" at the end must have an additional "s" placed after the apostrophe--Chris's not Chris'. If the possessive is plural, however, no additional "s" is required--the Carsons' house.
The only time you should put punctuation outside of quotation marks is when a semi-colon follows the quote, as in "She said the book was 'dog-eared'; I bought a new one instead."
Proper use of the comma is a common debate among scholars. Some argue that when it's used in a serial, no comma should follow the second to last item, while others say a comma should follow all but the last item--apples, bananas, and oranges. (This is called a serial comma.) The unquestionable rule is that once the choice is made, you should be consistent in its application.
The semi-colon is used to connect two otherwise separate sentences. It's also used in serial form when the items described have commas between them, as in "She brought fruits, vegetables and salad; beef, pork and chicken; and pies, cakes and cookies."
The dash is often misunderstood and maligned. It may be used instead of parentheses in a sentence: "The animal--a dog--ate the leftovers." It may also be used to indicate a sudden break in thought, as in "When did you--I thought we covered that topic."