As you try to leave Professor Oak's Lab, Blue will challenge you to your very first Pokémon battle! This is a tutorial battle, so don't worry too much. Blue will use his newly acquired Eevee. Use Pikachu's 'Thunder Shock' to deal super effective damage. This battle is a great way to learn the basics of combat.
After defeating Blue, head back into Professor Oak's Lab. He'll ask you to deliver a special item, 'Oak's Parcel', to a Pokémon Mart in Viridian City. This is your first official quest! Before you leave Pallet Town, visit your Mom. She'll heal your Pokémon for free anytime you visit, which is incredibly useful.