Elephants show signs of playfulness and curiosity in various ways, such as engaging in playful activities like splashing in water, tossing objects around, and engaging in physical interactions with each other or other animals. They may also display curiosity by investigating new objects or environments, using their trunks to touch and explore their surroundings, and displaying interest in novel stimuli. Additionally, elephants may exhibit behaviors like vocalizing, ear-flapping, and tail-wagging when they are in a playful or curious mood.