Howard,
Sometimes it's really not that important to freeze a moving subject with the faster shutter speed possible (1/4000 sec. in some cases). Panning or following the subject while exposing the picture can be accomplished using a relatively slow shutter speed. This technique results in a blurred background while keeping the moving subject in focus. You can do it at 1/125th of a sec. in normal daylight (also depends on how long or short the lens is). A long telephoto will need a faster shutter speed due to inherent camera shake. A tripod or monopod is invaluable to minimize camera shake.
Other than a fast shutter speed, of more importance is the shutter lag time. This is the time it takes for the camera to respond to your pressing the "shutter release" button. Obviously, the quicker the response, the better, otherwide one captures great photos of the background, sans subject matter as you've observed.
Most point-and-shoot cameras have a relatively lengthy shutter lag time. Most digital SLRs are fairly quick although some are slightly faster than others. The best dig. SLRs will rival a SLR film camera.
Another factor is how many "burst" pictures a camera can take. Again, the digital SLRs have better specs. And within those that can record many frames in a burst, there is the time it takes for all those images to be written to the memory card. During this time period, the camera must wait while the images are written before resuming regular operation (next shot). Again, some cameras are better in this respect than others.
There are so many specs to compare you'll become dizzy. I'd suggest concentrating on the features that best fit the type of photography you like to do, but also be aware of zoom range, lens/picture quality, ergonomics, battery life, build quality, etc. :-)
To begin your headache, here are just a few sites to visit for camera reports, comparisons and user comments.
http://www.dpreview.com/
http://www.steves-digicams.com/
http://www.imaging-resource.com/
Hope this helps,
Sherwood