Hi Jess!
First, forgive me my English but I will try to help you anyway
1. Black and white pixels based image. Usually I do it in two steps. Image>Mode>Grayscale and then Image>Mode>Bitmap. The result is exactly you are looking for. There are few another ways to get this. For example, if you are not worry about details in shadows you can use Image>Adjustment>Threshold function.
2. As for 5 colors based image. Do next Image>Mode>Indexed Color. In the Indexed Color dialog box set
Palette: Local (Perceptual or Adaptive)
Colors: 5
Forced: None (or Black and White)
the Transparency box must be unchecked
Dither: None (sure you can play with Diffusion Pattern or Noise, the result will be slightly different)
Original image
B&W
5 color based image (Dither: None)
5 color based image (Dither: Diffusion 75%)
Good Luck
Andrey