I say shoot on Manual for the first 3 times you go out with it. After that, you'll appreciate Aperture Priority more. 😉
I've never used the "P" setting, as that seems exactly like Auto, except I think you have control over whether you use flash or not.
I think "P" lets you program in a specific set of parameters that you can then recall quickly. For example I have my "P" set to bulb, so whenever I want to do a bulb exposure, instead of having to flip through a bunch of menus I just set it to "P" and go.
Sorry, wrong. "P" lets you select from a set of shutter speed and aperture values that all have the same EV or "Exposure Value". I like "P". I'll go to shoot and notice the camera has selected a shutter speed I don't want so I can select a faster shutter and the "P" feature will adjust the aperture for me.
There is another mode that collects a bunch of settings for easy recal but Nikon has used "P" for "Program" from before the digital era and even in the pre-autofocus era. I think "P" showed up on Nikons in the early 1980's in something like the FE body but my memory is bad.
I think P is different from camera to camera. My XTi P is used as basically automatic mode, except you can shoot in RAW instead of jpeg, you can turn the flash on and off, you can control the ISO, and you can change the exposure (although it decides the shutter speed and the aperture)