So I went to Lowes and loaded up with some boards and got to work. This whole project cost less than $40 and took one evening to install.
I wanted the wainscoting to be somewhat on the tall side so I got some 4 foot boards (1/4 inch thick) and had them all cut at the same time to 3 feet.
We spaced them about 20 inches apart:
Then put one all the way across and topped it off with a chair rail:
And after two coats of paint, this is the final product: