After you win three Gym Badges, you can go north through the Wild Area and enter the city of Hammerlocke.
When you first arrive in Hammerlocke, go north and there will be a cutscene.
Enter the Stadium
Go north into the Hammerlocke Stadium and there will be another cutscene.
Before you leave the stadium, talk to the Ball Guy to get a Level Ball, which works best if your active pokémon is a higher level than the one that you are trying to catch.
From the stadium entrance, there is a sparkle in the yard of the house to the east. Check on it to get a Nugget.
In the house east of the stadium, talk to the person near the fireplace to get a Soothe Bell.
To the east, check on the sparkle on the crenellated wall to get a Dire Hit.
East of that, go into the middle house and listen to the person's explanation about weather to get a Heat Rock. Talk to him tomorrow to get a Damp Rock, then the day after that to get an Icy Rock, then the day after that to get a Smooth Rock, and finally, talk to him the day after that to get a Utility Umbrella.
Go east and up the stairs. There is a Strawberry Sweet to the left of the stairs.
Go north from there and check on the sparkle to get a Rare Candy.
Go east and talk to the child and agree to help. You get the Old Letter.
Go east from there to find a pair of Wise Glasses.
Go back down the stairs, then go west and enter the Pokémon Center. The Poké Mart on the right sells Dive Balls, Dusk Balls, and Timer Balls.
Outside, enter the house to the left of the Pokémon Center, and you can talk to the trainer to battle Black Belt Brian, who has a level 2 Cottonee. The Cottonee is holding a Focus Sash. You get 128 Pokédollars for winning, as well as a Focus Sash.
When you are ready to proceed, go outside and go west. Approach Leon and there will be a cutscene.
Afterward, go west and you will find a clothing boutique, a Battle Café, and a hair salon. To the west of the hair salon, check in the corner to find a sparkle. Check on it to get a Super Repel.
Go north from there to find 2 Hyper Potions.
To the west, you will cross a bridge, and there is another Pokémon Center here. To the left of the Pokémon Center, check on the sparkle to get 2 X Attacks.
West of that, check the southeast corner of the park to find a Muscle Band.
If you talk to the person in the southwest corner of the park, you can trade a Toxel to receive Holly's level 25 Togepi, named Snips.
North of the park, Raihan is standing at the door to the Vault. Walk up to him and there will be a cutscene. You will receive Raihan's League Card.
Visit the Vault
After the cutscene, go up the stairs. When you are outside, go south, then go all the way east to find TM29 Charm.
Then go up the stairs and go through the door. Approach Sonia and there will be a cutscene. You will get 2 Revives.
Go back downstairs and there will be a cutscene.
Finish Exploring Hammerlocke
Go west from the Vault door and then go southwest and talk to the person standing on the grass. You can give the person there an Applin, and there will be a cutscene. You will get your Applin back, and you also get a Tart Apple.
Go southwest along the grass where the Applin person was, and talk to the person at the end of the path to teach your starter pokémon a battle-combo move.
Go north again, then go west across the drawbridge to reach Route 6.