After exiting Mt. Moon, you'll arrive on Route 4, which leads directly to Cerulean City. Heal your Pokémon at the Pokémon Center immediately. Cerulean City is home to the second Gym Leader, Misty, who specializes in Water-type Pokémon.
Water-type Pokémon are strong against Fire, Ground, and Rock types, but weak against Electric and Grass types. This is where your Pikachu will shine! Make sure Pikachu is well-trained and has learned 'Thunder Shock' or 'Quick Attack'.