Pokemon X and Y Walkthrough: Route 2
Your rival and Shauna are waiting to teach you how to catch Pokemon. In essence, you need to drain a Pokemon's health by attacking. Get it as close to 0 as possible without knocking them out. Then, throw a Poke Ball and cross your fingers. There are other factors that go into it, like status effects, but none that you have control over at the moment. After the lesson, your rival gives you some Poke Balls so you can go catch some Pokemon of your own.
Further north is the entrance to the Santalune Forest. There's a Youngster there waiting to battle.