Are you talking about shutter lag (the time it takes to take a picture once focused - shutter halfway down)? Or are you talking about the time it takes to autofocus (shutter halfway down) and then snap the picture (which is what it has to do if you aren't already focused and you just press the shutter all the way down)?
I suspect the problem is less the manual settings and more shutter lag/time it takes to autofocus. It doesn't take very long to meter a scene and auto expose it properly, though for sports it might be nice to be in shutter priority mode and dial up as much shutter as you can.
Honestly, if you're going to be shooting sports and are quasi-serious about getting good shots, you need to get an SLR. Even the entry level ones are miles ahead of the point and shoots. A decent SLR (you can get a Rebel XT on ebay for under 300 probably...it's not as gucci as the newer XSi and XS, but picture quality is every bit as good) and a reasonable USM zoom lens (the EF 100-300 4.5/5.6, for example which can be had for 200ish) would be a start. Some of the lenses alone for shooting sports can cost a few grand, so don't expect to be shooting covers for sports illustrated, but daylight shots at a reasonable ISO (400 or 800) will give you some good results with that combo, I'd say.
Sports and birding are the two most demanding tasks most people have for cameras, and people spend tens of thousands on setups for shooting those two things, so don't expect miracles, but you should at least be able to get some decent shots out of a cheap SLR setup.
Though you shouldn't have THAT much trouble with a point and shoot.