I want to animate a gradient node moving around from left to right, but I cannot figure out how to. I tried the Color Ramp and the Mapping node to control the gradient spline.
You can key-frame the location field(s) in the Mapping node.
At the moment you are mapping in Camera space, (i.e. the space of the whole rendered screen in XY, and Z in and out of it.) Maybe one of the spaces associated with your object would suit your purposes better, it depends what you want to do.
Select 'Texture' in the Mapping node, and right-click in any of the Location fields to set keyframes, and animate the position of the texture in X/Y/Z.