Reservoir: The most likely source of a whine noise due to a fine screen inside the reservoir getting dirty and restricting flow. Replace it or fill it to the top to see what happens to the noise.
Pump: The power steering pump is a pain to work on. Best to remove the wiper tray. Power steering pump can be a source of the whine as well.
Moving the wheels from underneath can actually cause spillage out the reservoir and induce air. Turn the wheels lock to lock a few times while driving in a parking lot should get rid of the air.
Serpentine belt is easy enough if you use a serpentine belt tool. Remove the splash shield to gain access.
Although many Mechanics think differently, or don't know, the power steering system calls for transmission fluid ATF+4.