After receiving the Pokédex, Professor Elm will call you on your Pokégear, urging you to return to his lab immediately. Head back through Route 30 and Cherrygrove City to New Bark Town. As you approach Professor Elm's Lab, you'll encounter a mysterious red-haired trainer attempting to steal one of Professor Elm's Pokémon. This is your rival!
He will challenge you to your first rival battle. His starter Pokémon will be the one that has a type advantage over yours. For example, if you chose Chikorita, he'll have Cyndaquil. If you chose Cyndaquil, he'll have Totodile. If you chose Totodile, he'll have Chikorita. This battle can be tough if your Pokémon isn't sufficiently leveled. Use Potions if necessary. After the battle, he'll flee, and you'll learn his name (which you get to choose!).