Route 5 is a short road to the south of Cerulean City. The route gate to the south will be blocked for now, so you’ll need to take the underground passage to the southeast.
The Pokemon Daycare can also be found on this route. You can leave one Pokemon here to level them up over time. However, they will not evolve, they will not get the same experience from battling (which boosts certain stats more), and they will automatically learn new moves without giving you the option to forget select moves.
When you’re ready to continue on, enter the small house in the southeast. This is the underground passage. The trainer inside will ask to trade Pokemon. The trainer will ask for a Cubone and give you a Machoke. You won’t be able to find a Cubone until much later, however.
Take the underground passage to Route 6 next. You may be able to find some hidden item on your way through the passage. Just keep tapping A as you walk.