If you don’t have a ticket, you need to go to Cerulean City and battle the trainers on the bridge. After you’ve defeated all of the trainers, head east and battle the trainers (and catch any wild Pokémon you come across). Once you’ve battled all of the trainers, go into the house. The Pokémon Researcher Bill will be there looking like a Pokémon. Go up and talk to him and help him out. When you have gotten Bill out of the Pokémon costume, he’ll thank you and give you the S. S. Anne Ferry ticket and ask you to go in his place.
If you don’t have a ticket, you need to go to Cerulean City and battle the trainers on the bridge. After you’ve defeated all of the trainers, head east and battle the trainers (and catch any wild Pokémon you come across). Once you’ve battled all of the trainers, go into the house. The Pokémon Researcher Bill will be there looking like a Pokémon. Go up and talk to him and help him out. When you have gotten Bill out of the Pokémon costume, he’ll thank you and give you the S. S. Anne Ferry ticket and ask you to go in his place.
You can find him by following the initial hallway south, then turning west and taking the stairs up at the end of that hallway. Once you’re up the staircase, follow the hallway south, then east, then north, and take the stairs at the end of that hallway. Those stairs will take you into the room the Captain is in. Talk to him to start the dialog where he gives you “HM01 Cut. "
Follow the path lined with bushes on one side and trees on the other east into the grass patches. Go north up the first set of steps, then head west and down the next set of steps, and follow the path west. Once you get to the sandy spot, head north to another set of steps, go east, and down the set of steps. Walk east to the tall grass and head north around the big dirt pile you were just on. Walk west around the large dirt pile to another sandy spot with a sign. Follow the path there west to the next zone. Walk west until you see a large patch of grass. Go up the second set of stairs that are located just north of the grass patch and go down the set of stairs on the west side of the plateau. Walk around the plateau to the west, into the tall grass. Head north until some bushes block the way. Walk east into the tall grass, then immediately turn west to bypass the blockade of bushes. Continue west into the tall grass and walk south to a pole-lined path that leads to the next zone. Walk toward the sign and head west past it until you see a building.
Talk to the Warden and he’ll reward you with “HM04 Strength. " Strength lets you move boulders in your way (such as caves, etc. ).
You’ll need HM01 Cut already. You’ll need to have at least 10 Pokémon in your Pokédex.
You’ll need HM03 Surf.
You’ll need HM03 Surf to continue north to the cave.