Quote History Quoted:
Priorities in my opinion
1) focus. If a picture is poorly focused, it is wasted. There is no fix for poorly focused images. A little soft is one thing. Blurry and unidentifiable is something else. Back focused etc. It'd also be good to mention here verifying what is in front of and behind your subject. Not only for compositional value but to make sure your camera doesn't jump focus on you.
2) light. With a point and shoot in particular, more is always better. Natural light for the win. Indirect or diffused like from a window or on a cloudy day. Bad light can be fixed to some extent in post but there are tradeoffs. The built in flash on your camera will cast some harsh light on your subject. Rig up or buy a diffuser for it if you have to use it. Sometimes it is useful.
3) composition. Easiest to fix in post but if you do it in the camera it saves time and trouble. How to learn? Study up on the rule of thirds and I would add in PHI. The divine ratio. Can be useful in composition also. Another thing go consider is look at pictures you like. Look at stuff from better photographers than you. See what they did. Ask how, if you arent sure. Ask why, too. Try to emulate some of the pictures you like.
You will be limited by the point and shoot but it is still possible to take great pictures with even a crappy camera if you do your part.
View Quote
I would like to throw something in here. Focus is VERY important, but focus on the eyes. If the forehead or nose are in focus, but the eyes are soft, it will look terrible.
A large aperture (low F stop number) will bout a nasty background, but make the area of focus very shallow.
Others have mentioned great books. There is also a lot of valuable free information on YouTube.
Do a youtube search for composition, rule of thirds, sunny sixteen rule (not that important with modern cameras) , aperture, shutter speed, and ISO.
If you are going to use flash, use an external one that will let you bounce the flash if possible. If not, use a diffuser.