First, remove the battery pack. Then, using the flat-head screwdriver, pry off the plastic panels on the controller's grips by wedging the screwdriver between the triggers and the plastic panels below them. The panels will pop off.
Remove the rubber joystick covers by popping them off like a Lego brick. Then, wiggle the metal joysticks underneath to loosen them.
If you find that there is literally sticky stuff holding them in place (if your house is like mine, it's Mtn Dew), use a damp cotton swab to clean away the stickiness and let it air dry. Reverse these instructions to put the controller back together.
If you're having button problems, continue to the next step.
Make sure to slide the second board gently into place over the top of the buttons so they don't roll out and into other parts of the casing. Also make sure the board slides under the charging port's plastic prongs.
Now, just put the controller back together in reverse order.