You want to start thinking of colours using a system known as HSV (Hue, Saturation and Value) - this will help you understand how colours work together.
Hue is the colour - e.g. red, blue, purple. Hues go in a circle through all the colours of the rainbow.
Saturation is how colourful the colour is. Saturated colours are rich and sometimes sickly. De-saturated colours are more subtle and subdued. If you remove all saturation from a colour, you end up with a pure grey.
Value is how bright the colour is, ranging from pure black to white, with shades in between.
Why do you care? Because almost all the 'rules' of good colour can be expressed easily in terms of hue, saturation and value. For example:
Adjacent hues complement each other well. Think colours in the rainbow that are next to each other:
Distant hues - when drawn on top of each other - usually look horrific:
Unless they're chosen carefully, with a balance of value and saturation. One trick is to make one colour much darker. Then they can be quite striking: