Most newer P&S cameras today have improved on focus speed and response speed over those from 4 or 5 years ago - though overall whether they are 'better' or not is subjective. But one thing some folks don't realize with cameras is that, despite their 'automatic' modes that make things seem so simple, a little photographic knowledge and simple technique skills, go a long way towards improving results.
With subject movement, any camera requires two things - a fast shutter speed, and lots of light. The fast shutter speed should be possible with any camera - even pocket cams should be capable of 1/1000 shutter speeds...light is where the issue comes in. Really bright, good daylight is the ideal, but in lieu of that, cameras will need larger apertures to see through, and will essentially have to 'turn up the gain' to be more sensitive to light. The aperture of a P&S camera is usually not going to be very wide, and the gain (in digital photography, this is the 'ISO' setting, using the old film speed numbers to denote more sensitivity) is something you don't want to turn up too much on compact cameras because with increased gain comes increased noise (just like on shortwave or CB radios of old). So ideally, in a new camera what you'd be looking for is a fairly wide aperture, denoted by the smaller 'F' stop number on the camera's specifications - minimum should be at least F2.8, with some compacts going down to F2.4 or even F2.0 (though usually these are also more expensive models)...while the larger sensor cameras (1/1.7" sensor) will always do a little better with higher ISOs than the smaller sensor cameras (1/2.5" sensor). None will be fantastic, but bigger is always a little better.
But the second topic is how you shoot. You noted a 1/2 second delay when you press the shutter until you get the shot. Are you referring to seeing the subject, pressing the shutter button all the way down, then waiting for the camera to focus, meter, and shoot? Or are you referring to seeing the subject, half-pressing the shutter to focus and meter, framing your shot the way you want while holding the half-press shutter, then pressing the shutter fully to snap the photo? The second technique is the best one to use for photography...most photographers use the half-press method when shooting, whereas many non-photography folks often use the mash-the-throttle method. Even old, slower cameras from 4 or 5 years ago, when using the half-press method, should have extremely short shutter delays, on the order of milliseconds...certainly no more than 1/8 second or so. But even current cameras, though a bit faster than your 6 year old model, will still experience a bit of a delay between pressing the shutter and taking the shot if you fully press the shutter all at once. Just something to consider, as you may be able to get better results even from the camera you have!
