An example for punctuation around speech marks and brackets could be:
He said, "Paul told me to go to the market and fetch his soup".
He said, "Paul told me to go to the market and fetch his soup."
With with brackets/parentheses:
At the market (which was only down the road.)
At the market (which was only down the road).
While an example for the use of apostrophes could be:
"The Rolling Stones' first chart single"
"The Rolling Stones first chart single"
Because phonetically there is no difference, many writers assume no apostrophe is needed, yet that is a mistake just as it would be if it were missing from a name:
"Ross coat is upstairs" is wrong; rather, it should be: "Ross' coat is upstairs"
Although the rules themselves are actually simple, they can trip up even established writers because there is no way of knowing what to do based on how it sounds. So, to make it simple, apostrophes go after the existing 's' (Rolling Stones) if it's possessive. A good way to remember it is by imagining a name without an 's' at the end and determining if it would need one in this context. For example, "Coldplays new album" would obviously need to be "Coldplay's new album". Thus, if it's a noun with an 's' at the end, add an apostrophe afterwards - "The Rolling Stones' new album". Simple, right?
It's no more complicated with punctuation with quote marks. Simply put, if the quote actually included it, add it within the quote; if not, add it outside. For example, if you quote a person to the end of their sentence, the full stop goes inside:
"Turn left at the top of the stairs for the bathroom."
If, however, it's not the end of the sentence, place it outside:
They told me to "turn left at the top of the stairs".
With brackets/parentheses, it's incredibly easy: if they are used within an existing sentence then the punctuation mark goes at the end, outside the bracket, otherwise the whole sentence is left wide open:
It started to rain (and Angie liked that).
If, on the other hand, the sentence ends prior to the use of brackets, the parentheses become their own sentence and so the full stop is placed inside:
It started to rain. (Angie always enjoyed the rain.)
That's pretty much all there is to it. The only exception is if you're using US English, in which case a full stop goes inside a quote mark even if it's technically incorrect - for example, if you list a song in quote marks (as you should), it would be thus:
The single that really made them famous was "Fun Time."
Even though, of course, the song title doesn't have a full stop in it, and placing it within quote marks would infer that it does. It's a strange anomaly with American English that a quote will hoover up punctuation that should come after, but, right or wrong, when in Rome...