Shutter delay is the biggest pain in consumer level digital cameras. If you look for a new camera, go to dpreview.com and read the individual reviews on the new cameras before buying. Phil usually talks about shutter delay in his reviews.
A digital SLR will have the least amount of shutter delay. The Canon Digital Rebel was mentioned and is probably one of the least expensive digital camera bodies that is available. It will accomplish exactly what you're looking for.
Now . . . that said, do I miss a point and shoot digicam? Heck yes . . . but I do LOVE my DSLR tremendously! I'd never get good shots of sports without it.
Lisa