Shutter speed (1/125, 1/500, etc.) won't effect quality.
Usually if you press the shutter half-way down, it looks at the scene and pre-calculates everything- focus, shutter speed, aperture, etc. and "locks it", waiting for you to hit the button.
Pressing the shutter the rest of the way actually takes the picture, but there may be a period of time from the time you press the shutter and the time it actually records the scene, otherwise known as "shutter lag". For instance, you may hit the shutter just as a diver hits the water, but it actually records it after he's halfway in. If you take a lot of action shots you get a feel for your camer's shutter lag and compensate for it. Naturally, if you don't pre-lock the scene and just hit the shutter, it has to do all the above (focus, calculate exposure, and record the scene) all at once which can take even more time.
None of the above effect the quality.