You should not attempt to put the car into P if it is moving at all as the transmission is physically locked; it is much more secure than the handbrake. N is the equivalent of neutral in a manual gearbox - I drive automatics and find that I never use neutral.
For 99% of driving there is no need to use anything other than P and R, although you may wish to manually change down a gear or two when going down a long hill, just as you would in a manual car. The method of doing that depends on the gearbox so you'll need to look at your handbook.