How to make custom textures?

I’ve made a simple model (party hat) and I’m trying to modify the texture, however as far as I can figure out (and look up online without getting utterly confused/frustrated) I can only put the color to be universal.

So how do I change the texture/make a new one? (I.e: Make the background of the hat red and add polka dots of various colors on it)

Texturing for Blender Internal: Realistic Orange

Texturing for Cycles: Texturing and Rendering a Shoe

You can also Google, search YouTube, this forum and Blender Cookie’s web site. Tons of stuff out there.