Sometimes you don't have time to take in a scene thoroughly.  This is especially true if your subject is capable of moving and thus changing the composition.  People, birds, animals and insects generally (but not always) fall into this category.
 

However, if I have time to immerse myself in a subject and think about the best ways to capture what is in front of me, I find it helpful to think in terms of threes - find at least three different ways of looking at the subject or changing the perspective (ie. not just altering the shutter speed or aperature, although changing those can be worth playing with as well).

Here's my general approach.  I'm using some images from Sayulita, Mexico as my examples:

My first inclination is to take an establishing shot, one that shows the entirety of the subject and all the possibilities therein.  Note:  In this case, this was the last shot I took because I kept trying to get as close as possible without disturbing the birds.  This photo was taken with a 20mm wide-angle lens and feet virtually in the water.  When I tried to take a step closer still, the splashing of my foot coming up scared the birds away.

The next step is to try and find interesting compositions or subjects in the scene.  In this case, I found the pair of pelicans towards the center of the scene appealing.  I liked the contrast of the one basking and the other not, the one with the more yellow in the head while the other more white.

Almost always, my final step is finding individual details that I find striking.  In this case it is obviously the individual birds and trying to create portraits of them.

I have lots of examles of these kinds of triples that I'll post in the coming week(s).  Or, see if you can find them yourselves :)