I know that I can change the color mode for an image but does that change it for the palette too. I can't see any change in the palette when changing the image color mode. It seems inaccurate this way, can anyone tell me what happens with palette color / how to change it?
No, it doesn't change the color picker. Instead, it just changes the way photoshop handles that color internally as numbers. So instead of RGB numerical values, it will store the color info as CMYK values.
It also affects the gamut, which is what colors a printer can actually print.