how do i remove bathtub stopper without making a mess?

Put a towel in the tub first so nothing small rolls down the drain. Then it depends on the type: a toe-touch one usually unscrews anticlockwise, a lift-and-turn has a small set screw under the knob, and if there's a lever on the overflow plate, the stopper is attached behind that - unscrew the plate and lift the linkage out.