Instructions
Travel to Vermilion City and enter the S.S Anne on the east side of the ship. You will need a ticket to board the ship, so if you have not already collected this item you will need to visit Bill on Route 24. Enter his cottage and talk to him to receive the ticket. Once you have the ticket, board the S.S. Anne.
Work your way through the boat, battling trainers along the way. If your Pokemon faints or become low on HP you can rest in one of the beds to recover. Eventually you will reach the Captain's Cabin.
Battle and defeat your rival and he will step aside, allowing you to enter the Captain's Cabin.
Walk up to the Captain and speak to him. Once the conversation has ended he will give you HM 01 Cut.
Press the "Start" button to bring up the in-game menu. Navigate to "Item" and select "HM 01" from the list.
Choose a Pokemon from your party who you want to teach the move Cut. Any Pokemon who are eligible to learn the move will be marked with the word 'Able" next to their name. Once you have selected a Pokemon you will have to confirm that you want to teach them the move. Once a Pokemon has learned a HM, the move can not be removed.
Walk up to a small shrub in the game and press the "A" button, then press "Yes" in the dialogue box to cut down the shrub. This move can only be performed when a Pokemon that knows the move Cut is in your party.