After you go through Route 25 and help Bill, you can go south through Cerulean City to reach Route 5.
Explore Route 5
After you get TM10 Dig, you can go south from Cerulean City to Route 5. If you want to go into the grass for a chance to catch wild pokémon and find items, jump down the ledge in the area south of Cerulean City. Otherwise, go south down one of the paths next to the ledge for a shortcut.
If you go into the grass, you can find Vulpix, Abra, Pidgey, Rattata, and others.
On one of the ledges, you find 3 Great Balls.
Check the bush to the left of the building to find 3 Razz Berries.
Go into the house to find the Pokémon Day Care. You can leave one pokémon here and it will earn EXP while you are away. The more you travel while your pokémon is in Day Care, the more EXP it will earn. When you pick it up from the Day Care Lady later, you have to pay a fee. The more levels your pokémon has gained, the higher the fee will be. There is also a chance that your pokémon will pick up items while in Day Care, such as berries.
Go southwest from the Day Care to find an Ether.
Go east from there to find 3 Nanab Berries.
Go south from there and your rival will talk to you. During the cut scene, you will get 3 Revives and you will give one of your S.S. Tickets to your rival.
There is a gate house south of the Day Care, but you can't go through it right now. Instead, go into the Underground Path where your rival went. Go down the stairs to enter the Underground Path.
Go south and you will find a Repel.
Keep an eye on your partner pokémon's tail as you go through the Underground Path. There are many hidden items here. If your pokémon's tail is wiggling quickly, you are standing near a hidden item, so check the ground around you to find it. In the Underground Path you can find hidden items such as Nuggets, Big Pearls, Lures, Fresh Water, Super Potions, and more. New hidden items can appear over time.
Farther south, you find a Lure.
At the south end of the Underground Path, go up the stairs, then go south. You will be in Route 6.