How to Proceed

I am just learning this game and have a question about openings.  If I'm playing white and black doesn't reply with the standard moves to my opening, should I continue with my standard opening moves or adjust them to what black does?

It depends on the position. The most difficult thing to learn about openings for me, was only getting to the second or third move before my opponent wouldn't follow the opening anymore. I eventually gave up on following openings.

Play the board, and position. If you do not, you could get in real trouble fast. Many players know of obscure moves not necessarily covered by standard book, or its a transposition, that will favor them. Don't just blindly follow a standard opening or set of prescribed moves. If you lose the game, go through it carefully, and check all sources to see if it was a transposed move or out of book, which should have been listed in your opening book. 

In the opening stage, you have to do mainly 4 things:
- Control the center
- Develop pieces (minor pieces, knights, bishops)
- Make castling
- And finally move the queen, to connect the rooks.

So, if your opponent plays a non-book move, then you still can play a very good game. Or even if your opponent plays an opening what you don't know, you do not need to worry, just follow the usual openings principles.

This is not advice because I am not qualified to give it but this is what I am now doing.

If in the opening my opponent does something out of the ordinary  which most of the time is an aggressive move for one of my pieces, I think why are they doing this and what in doing it have they left open.

I think opponents do this for one main reason, to intimidate, to try an halt your development in the opening.  I now tend to develop despite the threats...in a measured but aggressive way.

You get nothing for free in life, so if they wants something of mine, what of theirs is on offer, even if that unwilling trade is just to make them move their King so they cannot Castle.

Many thanks to all who replied! As a beginner, there is a great deal to be aware of. Concentrating on development principles is the correct course of action for me at this point in time.

Certainly NOT continue as if nothing happened.

You have to adapt. If your opponent deviates from the standard moves, it must be wrong. If you don't know why it is wrong, you have learned to much of the opening.