Phillip is pretty much right on the money here.
Depending on what you are shooting, you will more often than not need to control either of aperture or shutter speed, and let the camera take care of the other to get your exposure right.
Most of the time, I will use aperture priority, set at a "safe" value of between 5.6 and 8. I keep an eye on the shutter speed and open up the aperture or bump up the iso if it starts to dip under 1/f. If I want to take a portrait shot or minimise my depth of field, I will open the aperture up and let the camera deal with the shutter speed.
If shutter speed becomes my primary concern, then I will use shutter priority... For example if I am shooting sports, I know I want it very fast, so I will set it accordingly and let the camera adjust the aperture.
If I'm going to be using flash or taking a long exposure on a tripod, I will probably have it on manual.
In fact, even when you are using manual, you will probably use it very similarly to A or S
modes. ie: you set one of them to what you want, then you adjust the other until your exposure is right. The only difference is that in A or S
mode, the camera does that second part for you automatically.