Wellspring Cave is a basic cave with two floors, northwest of Route 3. A small section of the first floor can be explored to begin with; HM03Surf is required to delve deeper. TM70Flash is also advisable, as the second floor is pitch-black.
You first visit this small cave to chase after Team Plasma, who have hidden here after stealing a girl's Pokémon. Cheren also gives chase, and is facing up to Team Plasma when you enter. Team Plasma try to justify stealing the Pokémon, saying that a Pokémon cannot reach it's full potential under the guidance of a child, and challenge you to a battle to try to prove their point.
After beating one grunt, more will come to assist, resulting in the game's first tag battle. Cheren aids you with his starter at Lv. 14. Team Plasma return the stolen Pokémon once they're defeated, however they reveal their plans to steal more Pokémon to liberate them from humans.