The first thing you'll see in Route 3 is a set of stairs. If you walk up the stairs, you will see two buildings. The building to the right is the Pokemon Day Care, where you can drop off your Pokemon for two reasons. One reason is to breed two compatible Pokemon which results in an egg you can hatch. The other reason is to have the Pokemon gain a certain amount of experience for each step you take when it is in the Day Care. The building on the left is a preschool. You can go inside this building and talk to the person just in front of the doorway and she will heal your Pokemon so you won't have to walk all the way back to Striaton City.
West of the preschool is a sand box with a hidden item, but first you must battle a Nursery Aide. She has a Munna that can put your Pokemon to sleep, which can be very annoying. It's always good to keep Full Heals and Awakenings on hand for such an occasion.
Walk over to the sand box. You will find a hidden Ultra Ball if you walk to the top-left corner of the sand box, take one step right, take two steps down, and press A. Except for the Master Ball, Ultra Balls are the best non-specialty balls to catch Pokemon. The next trainer to battle is next to stairs of the slide.
Heal up inside the preschool and make your way back down the steps and west where you will fight your first double battle. If you are unfamiliar with double battles, it is when each team has two Pokemon active at a time. The two Pokemon of yours that are chosen are the top two slots in your party. Make sure you have the two Pokemon you want to battle at the top of your party by pressing X, selecting the Pokemon option, and switching your choices to the top of your party.
Keep walking west and you'll see an item in the patch of grass. Don't get too excited because Cheren will show up to battle. It shouldn't be a very tough battle now that you have the elemental monkey that can take down his starter Pokemon. His starter Pokemon holds an Oran Berry, so when it gets down to below 33% health, it will replenish 10 HP.
After the battle, two Plasma Grunts will run past followed by Bianca and a little girl. Team Plasma stole the girl's Pokemon so you and Cheren need to get them back. After everyone is done talking, Cheren runs off west to chase down the Plasma Grunts. You can now get the item ball in the patch of grass, which contains a Super Potion. If you try to go south, Bianca will stop you. You can still go back to the preschool to heal you're Pokemon if you need. Head west when you're ready.
Walk west until you hit a wall and then go north until you see Cheren. If you talk to him he will ask if you want to go into the cave together to track down the Plasma Grunts. Say no for now because there is still a trainer to fight and an item to find if you walk east through the grass. First, you'll battle a Pokemon Breeder.
She will give you an Oran Berry after you defeat her. Walk south from the Pokemon Breeder to find a Repel. Hop down the ledge and make your way back to Cheren. Talk to Cheren and say yes to his question. He will go into the cave and you need to follow him.