Ok, well opinions on this one vary - a lot. As for my own method, generally I find that I get my best results overall using A mode (Aperture Priority) - I can set the Aperture so that I get my desired depth of field, then adjust my ISO so that my shutter speed gets set high enough to get the sort of effect I want. I prefer a high shutter speed, my preference is for shots that are "frozen" and have little to no motion blur. So in practical terms I generally adjust my ISO so I have something like 1/800 shutter speed or higher for large critters, 1/1250 or higher for smaller, faster critters.
There are times, particularly when I'm shooting smaller, faster animals and particularly birds and the light level is varying a lot that I'll switch to S mode and that way I can be certain my shutter speed stays high enough regardless of lighting, it's a lot easier to fix exposure in post process whereas a shot ruined by motion blur is most likely unfixeable.