All digital point and shoot cameras share that annoying problem known as shutter lag or delay.

No matter if you press the button half-way down to focus first,when you finally shoot, 'ol Rover is already out of the frame.
The answer?a digital SLR of course,but not ALL digital SLRs are as rapid as you like.

They are,of course,more expensive such as a really fast shooting camera like the Nikon D-80 will set you back nearly a grand with accessories.

Do a google search with the keywords shutter delay times and you'll get a representative selection of tables indicating the shutter speed and frames per second of the leading SLRs
Ahh! Here we are.

http://www.impulseadventure.com/photo/shutter-lag.html