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You'll find this section toward the south end of Vermilion City. A guard will stop you if you try to walk onto the walkway. Press A to do so. The guard will allow you on-board at this point. Board the boat and make your way through it, battling anyone you encounter when prompted. You'll ultimately run into your rival here; after defeating him, you'll be able to continue along to find the captain. Head up the stairs behind your rival to enter the captain's quarters. You'll find him next to a bucket. Walk up to the captain and press A to do so. Once the back rub is complete, the captain will reward you with the "Cut" HM. You can now use it with your preferred pokémon. Once you've allowed one of your pokémon to learn the "Cut" HM, you'll be able to cut down path-blocking trees throughout the world by selecting the pokémon and using it on the tree(s).
Head over to the boat walkway. Show your pass to the guard. Defeat any trainers and your rival. Find the captain. Rub the captain's back. Receive the "Cut" HM. Apply the "Cut" HM to a pokémon.