I'm planning on making tons of GIFs / animated images, but would like to know where to start.
That is not a GIF, that is an MP4 file, but ok, that is a minor detail.
You need a program to draw the elements, a person, the screen, the shadow. Use a vector-based program.
Inkscape => Free
A program to animate the elements in a loop.
Blender => Free
NCH Animation Express => Free
Hit Film Express => Free
Probably a program to convert the rendered video into an animated gif.