Here are the proper movements to accompany this finger play:
"The itsy, bitsy spider climbed up the water spout": Place your right thumb on your left forefinger, and your left thumb on your right forefinger. Release the right thumb/left forefinger and twist your hands so that they rejoin above the left thumb/right forefinger. Repeat this motion so that your hands are "climbing" upwards.
"Down came the rain and washed the spider out": Raise both hands in the air, fingers splayed, and whoosh them down in a "rain" motion. Then, when your hands are down near your belly, sweep both hands outward, away from your center.
"Out came the sun and dried up all the rain": With your hands at your waist, palms facing forward, move your hands from side to side and upward at the same time.
"And the itsy bitsy spider climbed up the spout again.": Repeat the first "climbing" motion.