When you go through Cherrygrove, you should stop in at the pokémon center to heal up. Then, as you're heading for the exit of town, the red head guy who was hanging out around Elm's laboratory stops you and battles you.
He will have one of the three starter pokémon that Professor Elm offered to you. In fact, he has the pokémon that is strong against whichever one you picked. He won't be that hard, but don't let your guard down.
When the battle is over, you'll see him drop his trainer card.
Continue to New Bark Town. You can use ledge jumping to avoid any fights. Once you're back in town, go into the laboratory. Elm will say that one of the two remaining starter pokémon was stolen. You know who did it! The police officer will suspect you of being the culprit, but your friend Ethan or Lyra will come in and clear your name.
The police officer will ask for the thief's name. You can make up a name, but the default name is Silver, so I'll refer to him as that.
After the police questioning, you will show Professor Elm the egg, and he'll be pretty surprised. He will also see that Professor Oak gave you a Pokédex, which must mean that he saw great potential in you to be a trainer. He says that you should set out on the Pokémon League challenge, winning badges from all of the pokémon gyms in Johto. He tells you to go to Violet City to start your adventure.
Before you do, go talk to your Mom. She offers to save some of the money that you win from each trainer battle. This is a good idea, because sometimes, after a battle, she will use some of your savings to buy some rare items for you. You can always call her and ask her not to save money, if you decide that you would rather have it in your pocket. After all, you would have to go all the way back to home to retrieve your savings. But those rare items are worth it, so give it a shot.
Now, head out and go toward Cherrygrove City again.