Upon arriving in Agate Village, you'll be introduced to the Elder, who explains the significance of the Celebi Shrine. This shrine is where you can fully purify Shadow Pokémon once their Heart Gauge is completely empty. The Elder also introduces you to the concept of 'massaging' your Shadow Pokémon to speed up the purification process. This can be done by a specific NPC in the village.
Take advantage of Agate Village's resources. Heal your Pokémon, stock up on items, and most importantly, begin the purification process for your snagged Shadow Pokémon. Having a diverse team of purified Pokémon will be essential for later challenges. Remember that purified Pokémon gain access to their full move pool and can level up normally, making them much more effective in battle.