Digital camera question - shutter speed/frames per second

[NewJeffCT]

A few years ago, we bought a pretty good Sony digital camera - 3.2 megapixels with 3x optical zoom. However, one of the disappointments we had is that it does not take pictures quickly. (even worse if the flash is used!) So, if we take one picture, it will often take a few seconds before the camera is ready to take a 2nd pic... and, if you're indoors with the flash, it could be several seconds before we can take a 2nd picture. With a very active almost 3 year old, we can lose out on a lot of good pictures this way, as you often lost the moment in a few seconds.

so, my question is, what should I look for if we want a digital camera that can take pictures at a faster pace? Is it frames per second? Or, is it shutter speed?

Case in point - this Fuji camera Fujifilm FinePix E550 is: Max shutter speed 1/2000 sec Min shutter speed 3 sec according to CNET

While this slightly more expensive Canon Canon PowerShot A620s Max shutter speed 1/2000 sec Min shutter speed 15 sec

What is better for minimum shutter speed? Or, am I totally off base in my thinking?

Or, do I look at something like continuous shooting speed - the Canon is 1.9 frames per second...the fuji camera didn't seem to list that.