My husband left his account open so I am using it to ask for advice because he tells me ARFCOM knows EVERYTHING. I have a Kodak 30x IS Point and Shoot Camera (I know it's not that fancy but it's what I have). Anyways, I would love to take good photos of my kids (like the ones you could blow up and put on your wall good). The only problem is is that I am not good at all. I have read the manual to my camera a few times, have practised outside, read some blogs and my pics are still pretty crappy. I really would love your advice on where to start. I know it takes a lot of practice but definitely need suggestions on where to start with a point and shoot camera and I am welcome to any advice that has helped you along the way.

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.

Thanks so much for the advice. I will definitely do what you said and hopefully through study and practice I will get better. I would like to take portraits of my own family instead of having someone else do it. Is there any books, blogs, or etc that any of you have found useful? My camera has 12 mega pixels, how big of a picture can I blow up without it getting pixelated?

9x14 with no/little cropping.
8x10 with easy room for cropping.

Composition makes or breaks.  Be picky about your background; nothing ruins a good photo like a nasty background.  Use your feet rather than zoom whenever possible.  Google the "rule of thirds."  

The kit doesn't make the photographer. For instance, I've seen some really good photos taken with an iPhone.  

A better camera might help you get better pictures, but without knowing some basics of photography you could end up spending a lot of money for pictures you still aren't satisfied with. The camera's user guide will tell you how to manipulate the camera's controls, but is unlikely to provide anything but the most basic of photography tips.

Consider picking up a copy of Bryan Peterson's "Understanding Exposure" and/or Tony Northrup's "How to Create Stunning Digital Photography". These books will walk you through the concepts of composition, aperture, shutter, and ISO to help you make the most of whatever camera equipment you have.

Another option is to post some pictures you've taken along with what camera settings were used and some helpful folks can offer suggestions for how to improve them.

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.