After crossing Nugget Bridge and dealing with a Team Rocket Grunt, you go east to Route 25.
Explore the Route
Go east in the route until you see a Hiker up above. Talk to him to battle him. Hiker Franklin has a level 14 Onix. You get 560 Pokédollars and 3 Poké Balls for winning.
In the grass in this area, you can encounter Psyduck, Bellsprout, Pidgey, and more.
Go west from the Hiker and you will find an X Sp. Def.
Go back to the east and you can battle Bird Keeper Joey, who has a level 15 Pidgeotto. You get 900 Pokédollars and 3 Poké Balls for winning.
South of the Bird Keeper, you can battle Youngster Chad, who has a level 12 Diglett. You get 240 Pokédollars and 3 Poké Balls for winning.
To the left of Chad, there is a Super Potion.
Go east and you can battle Youngster Dan, who has a level 12 Ekans and a level 12 Rattata. You get 240 Pokédollars and 3 Poké Balls for winning.
South of Dan, you can battle Hiker Nob, who has a level 14 Geodude and a level 14 Machop. You get 560 Pokédollars and 3 Poké Balls for winning.
Go east and you can battle Picnicker Kelsey, who has a level 13 Goldeen. You get 260 Pokédollars and 2 Great Balls for winning.
North of Kelsey, you can battle Camper Dustin, who has a level 13 Squirtle. You get 260 Pokédollars and 2 Great Balls for winning.
Go east, then south, to find an Ether.
East of there, you can battle Lass Haley, who has a level 12 Oddish and a level 12 Pidgey. You get 240 Pokédollars and 3 Poké Balls for winning.
Go east and you can battle Fisherman Wayne, who has a level 14 Magikarp and a level 14 Krabby. You get 560 Pokédollars and 3 Poké Balls for winning.
Meet Bill
Go east and go into the house. You will meet Bill. Agree to help him. Go check on the PC, and after a cutscene, you will meet Bill again. Bill gives you some S.S. Tickets that you can use to board the S.S. Anne in Vermilion City.
Go back to Cerulean City
Leave Bill's house, then go east and talk to the Coach Trainer if you want to battle. Coach Trainer Amala has a level 16 Machop. You get 1600 Pokédollars for winning. After the battle, Amala gives you TM15 Seismic Toss.
Go south from Amala to find a Lure.
Go west from there and check in the bush south of the Fisherman to find 3 Nanab Berries.
Go west from there, then south, and check on the bush to find 3 Pinap Berries.
Keep going south to return to Cerulean City.
Back in Cerulean City
When you arrive, the police officer guarding the house will run after a pokémon that stole a poké ball. Go into the house. After the cutscene, go through the hole in the wall on the right.
Outside, you can battle the Team Rocket Grunt, who has a level 15 Raticate. You get 420 Pokédollars for winning. After the battle, the Grunt gives you TM10 Dig.
Go south from here, and keep going south down the path that you couldn't reach before. When you can't go farther south, go west until you find a Burn Heal.
Now you can go south to Route 5.