From Cerulean City, head south to Route 5. You'll find an Underground Path that connects Route 5 to Route 6, bypassing Saffron City for now. This path is straightforward and has no wild Pokémon or trainers. It's a quick way to reach Vermilion City.
Upon arriving in Vermilion City, you'll notice the S.S. Anne, a luxury cruise ship, docked in the harbor. You'll need a special item, the S.S. Ticket, to board it. A sailor near the ship will give you the ticket if you talk to him. Boarding the S.S. Anne is crucial for obtaining the 'Cut' HM, which is required to access Lt. Surge's Gym.