So I am watching the "Blender Beginners UV Unwrapping Tutorial", but cannot see how to add a texture to my material like in the color bar in the video (there should be a dot to click on). I remember having done this before. Has the current version of blender changed so much or am I missing something?
You are using the wrong render engine. You want to be using cycles, but you are using Blender Render. At the top of the screen there is an option to change the engine. You can see in the first screenshot where it says Cycles Render.