I usually play chess in my free time and have not really done any serious studying of the game, as I'm sure you can tell by my rating but I'm wondering how you make an opponent pay for making poor opening moves?  Are you just supposed to keep playing the opening as you know it or do you focus on their poor moves?

You dont follow opening principles.

You are hanigng pieces.

You are missing simple tactics.

Forget about trying to "punish" people, and stick to the basics to improve.  

The basics of each phase of the game

Opening:

Follow the Opening principles:

1.      Control the center squares – d4-e4-d5-e5

2.      Develop your minor pieces toward the center – piece activity is the key

Ø  Complete your development before moving a piece twice or starting an attack.

Ø  Move pieces not pawns.

3.      Castle

4.      Connect your rooks

Ø    By move 12, you should have connected your Rooks, or be about to do so.

Middle game:

When you have completed the Opening Principles, you are now at the middle game.  Now you need to formulate a middle game plan.  The middle game is a very complicated part of a chess game.  A simple way to develop a middle game plan is to perform the following steps.

1.      Scan your opponents 5^th, and 6^th ranks (3^rd, and 4^th if your black)

2.      Look for weak pawns, and or weak squares.

Ø  Weak pawns and squares are Pawns, and squares that cannot be defended by another Pawn.

Ø  Knights are excellent pieces on weak squares.

Ø  When deciding on weak squares, and weak Pawns to attack, the closer to the center the better

End game:

Start with the basics:

1.      Learn basic mates – KQ vs. K, KR vs. K, KRR vs. K

2.      Learn Opposition, and Key Squares

3.      Learn basic King and Pawn endings

Pre Move Checklist

1. Make sure all your pieces are safe.

2. Look for forcing moves: Checks, captures, threats. You want to look at ALL forcing moves (even the bad ones) this will force you look at, and see the entire board.

3. If there are no forcing moves, you then want to remove any of your opponent’s pieces from your side of the board.

4. If your opponent doesn’t have any of his pieces on your side of the board, then you want to improve the position of your least active piece.

5. After each move by your opponent, ask yourself: "What is my opponent trying to do?"

It depends on how "poor" these moves were.

If they hung a piece for no compensation, then you make the opponent pay by simply taking the free piece.  After that, you're out of the opening and you just play good chess.

If they simply didn't follow the book line (that you know), then there may be no immediate refutation.  Try to determine why the move you know is supposed to be the correct move and see how your opponent's move doesn't qualify.  For example, maybe the opening's play is centered around preparing/discouraging a certain pawn break.  If the normal move tries to prevent that pawn move, but your opponent's move ignores it, then the correct move for you may be to make that pawn break immediately.  Then again, maybe your opponent's move made some other threat that needs to be dealt with before you make the pawn break.

Perhaps if you show some examples from your own play, we can be more specific...

How to punish opponent for making bad opening moves?

Play better in the opening and middle game...that's the only way I know of!

It depends completely on what type of mistake it was. Off the top of my head, here are a few examples to show what I mean:

(1) Queen developed too early: Make him lose tempi by chasing the queen around with lesser valued units.

(2) Knight moved to B3 with no central pawns to repress the opponent's pawn advances: advance the center pawn(s) to make him lose tempi by retreating the knight.

(3) Bishop blocking a central, unadvanced pawn: Just play normally and he'll have to lose a tempo eventually to get that bishop to a useful square. Another option: try to advance your center pawn to your 5th rank to fork that bishop with the knight 2 squares to the side (if such a knight exists).

(4) Delayed castling too long in an open position: after your own king is secure, prevent him from castling, even if it requires a pawn sacrifice, then attack the stranded king in the center with the newly opened lines.

(5) Unsound minor piece for pawn sacrifice (usually at KB2): castle by hand, develop as quickly as possible, keep trading down pieces, and win the endgame from having 2 extra material points.

(6) Bc4 in the Sicilian: drive away the bishop with either ...a6 followed by ...b5, or ...d5, so that White loses a tempo retreating that bishop.

(7) Lost tempi: If 2 or more lost tempi then start looking carefully for a winning combination at each move since in all likelihood it exists or soon will exist.

(8) P-KB3 to protect the e-pawn: See if sacrificing your knight for the e-pawn (Damiano's Defense style) will give you a winning attack with Q-R5+.

etc.

See? Not much overall pattern there, and there are many punishment plans to know.

Chess is a hard game. To win in chess, all you have to do is punish your opponents mistakes, and not make mistakes. 

Chess is a fairy tale of 1001 blunders and Chess is a struggle against one's own errors. - Savielly Tartakower

I agree. For example in my black repertoire book, the authors say that you only have to learn the short "line":

"Develop and look for the weak squares on whites kingside."

A player could be lost by move six, however a lost position could be a checkmate by move ten, or being down a pawn that needs an endgame involving 50-70 moves to play out.  I'd say better to replace the terms "pay" or "punish", with "how to convert the type of winning advantage I have"

Why do people drag their queen out on move 2? Any threat she can make is easily countered, then she's chased around by knights. (mostly by knights.)  Maybe because my rating is so low a player rated 1400 - 1500 thinks a 5 move mate or something.  Terrible as I am, I'v never done that. Just hang around until I drop my queen. I do that at least once a day.

If the opponent is making weak, passive moves, try to make moves that threaten.  Don't abandon developing your pieces but don't be locked into placing them where you originally planned. Be aggressive and relentlessly point your pieces at your opponent's King whenever your opponent gives you the initiative. Carpe diem!

Play the position you have, not the position you want.