My first foray into routing out material was putting in a JLF and buttons, so dont let it scare you, its not that bad.
What I did, was first drill the through hole that the shaft will be using. Then you can insert the joystick, and trace the base in pencil on the underside of the wood. If you need a little extra notch for the connection (I didnt routing out 1/2" depth into 3/4" stock) you can trace that as well.
Then with a
Straight Mortising Bit set to the desired depth, start in the center of your hole and remove material in a circular manner. Then watch closely as you get near your pencil marks, and try to just trace out the outline with your cuts.
Once satisfied, try dropping your joystick in, if it fits, good work! If not, then carefully remove material until it does.