Walkthrough
Whether you choose to be wise and head north of Cerulean City to Route 24 in order to prepare to face the Cerulean Gym or choose to do so after defeating the Gym, you will encounter your Rival directly before stepping onto Route 24. He will tell you about a talking Pokemon on Route 25 - connected to Route 24 - before challenging you to another battle. He has added another Pokemon to his team, as well as leveled up the two Pokemon he had previously.
Pokemon Trainer Rival
Level 12 Pidgey: Quick Attack, Sand Attack, Gust
Level 12 Oddish: Absorb, Acid, Poison Powder
Level 13 Pikachu: Thunder Shock, Double Team, Quick Attack