I need a quick click... grand kids don't stand still....

The link that you gave me is Excellent. My current Sony DSC-100 apparently has a .3to.6sec lag (on Full Auto Mode). That is now the benchmark that I am shopping to replace with a Much Lower 'shutter lag'.

A BIG thanks for your answer; your link; and your speed in reacting.
A B I G THANKS!

A large part of the reason for the delay between when you press the shutter and when it trips is the Auto Focus (AF). Some cameras focus faster than others, true. But there's an easy way around that. If you're taking photos of kids, they are most likely around the same distance from you, just moving around. Point the camera toward the kids, and press the shutter button down half-way. This will focus the camera on the kid - if you have a focus box on your viewfinder it should turn green or bold to tell you that you have focus. Then keep the button depressed until you find your moment - that tip of the chin, thrust out tongue, hug or kiss - and press the shutter button the rest of the way. It will trigger instantly as you press it!
There are only two times this will not work well for you:
1. when you are shooting with a very shallow depth-of-field (something most point and shoots don't do very well anyway)
2. when the kids are moving toward or away from you for a great distance. But because most of these cameras have a wide focal range, you should get them in focus 90% of the time with this technique!
Good luck!

Another thing that might cause a camera to be slow? Poor low light performance. I like to take photos inside without using flash and I found that I was constantly frustrated with the blurry shots I'd get with my first digital, which was a Sony as well.

I searched all the camera sites like dpreview and looked for people discussing point and shoots with strong low-light performance and after reading many reviews, I decided on the Fuji Finepix F10, which is an amazing camera to this day, even though it is 4 years old. It takes very sharp photos in all kinds of lighting conditions. Last year I got the Canon S90 point and shoot which is a more advanced camera than the F10 but also takes great shots in low light situations. (Oh, and I have my DSLR, the Canon 5D, that is superfast, especially with a good prime lens, but that's another can of worms. )

I always keep the AutoFocus Assist beam ON, by the way, which illuminates objects that may be in dim light so the camera can get a strong focus. Sometimes this setting is not on by default so you'd have to check the menu of your camera.

Another thing you can do, even with your new camera is use a SD card with a fast write speed. The faster the camera can write the picture to the card the quicker it will be ready for the next shot.

Standard SD cards have a class 2-4 write speed capability.
Newer cameras like a class 6 or higher write speed (check your operators manual for suggested class/speed).

The higher speed cards will have a number on them showing the class/speed and will cost more as the class/speed increases.
This is true with the different type cards, ie SD,SDHC,Compact Flash etc.

What great ideas. Thanks. I am trying (at least eval'g) all that have been given. I appreciate the education.

The video will be the foolproof mode.
The reason the snap shots are good is that the preschools enjoy just sitting and flipping thru the photo albums we have made for them. They cant explain their fascination but always seem to sit quietly for those 'studious moments'.
My main reason to Still Photo is that I want to begin pencil drawing their young faces and am trying to use my photos to help me study the light & shade on features and unique expressions. Getting them to look up and hold the pose is thee challenge. Thanks again to all.

This is an interesting use of photography. Most people in photography will try to control and create the lighting they want. Many like to remove shadows, but shadows are not always bad. One can create some dramatic shots with carefully placed shadows. Knowing where to place the shadows is an art, and often distinguishes a master from an amateur. The more basic lightings are the broad/short lighting. More advanced ones may want to recreate the more dramatic Rembrandt lighting in the studio with 2 or more strobes. Studying portrait paintings and photos from old masters often help to understand lighting effect better.
So if you are mainly interested in getting portrait snapshots of your grandchildren for your portrait drawing, then you may want to consider lighting effect and equipment rather than buying a new camera. Studying lighting effect is an essential part of photography and drawing/painting. You need to know how to make use of available light, how to enhance it, and when to use strobes/gels to create more aesthetic lighting effect.
I have a number of lighting accessories to help me get the lighting effect I want at home, including a couple of hot shoe flash units, a set of studio strobes, remote controls for the strobes, gels, grids, reflectors, softboxes, umbrellas, beauty dish, LED lights, etc. I usually set up the lighting equipment with a dummy, make sure I got the lighting ratios right, then get my kids to where I want and interact with them, or let them play there. When the right moment comes, I activate the lighting and snap the shot. I almost never use hot constant lights for kid shots, they are dangerous. Lighting equipment can be quite expensive, but can enhance your portrait photos much more than a new camera and a new lens when done right.

The astore is a store that anyone, and that means anyone, can set up to sell any product that Amazon sells. I'd only buy from Amazon or one of the reputable third party affiliates(such as Cameta, Adorama, etc). They also have the zstore, which is for the higher sellers and I'd still not choose them, let alone the astore.