After defeating Brawly and getting the Knuckle Badge from the Dewford Gym, you can use Flash outside of battle and explore Granite Cave.
Catch Pokémon
In Route 106, which you pass through on the way to Granite Cave, you can use the Old Rod to encounter Tentacool and Magikarp.
Reach Granite Cave
Granite Cave is northwest of Dewford Town. You will go through Route 106 to get there. You'll pass by a couple of fishermen: Fisherman Ned, who has a level 12 Tentacool (level 11 in Emerald), and Fisherman Elliot, who has two level 11 Magikarp (level 10 in Emerald) and one level 8 Tentacool (level 7 in Emerald).
Teach Flash to a Pokémon
Inside of the entrance to Granite Cave, talk to the person standing there. He gives you HM05 Flash. It lights up dark rooms. The lower floors of Granite Cave are dark, so you will have to teach Flash to one of your pokémon if you want to see where you are going. If you use Flash in battle, it reduces your opponent's accuracy. Teach it to whichever pokémon you want to.
Granite Cave 1F
On the first floor of Granite Cave, you can encounter Zubat, Abra, Geodude, and Makuhita. Geodude works well in the next Gym, so I recommend that you catch one.
From the entrance, go west. In the northwest corner, you find an Escape Rope.
Go south from there and climb down the ladder.
Granite Cave B1F
You'll be in a dark room, so teach Flash to one of your pokémon if you haven't already, then use it to light up the room.
In this area, you can find Zubat, Abra, Makuhita, and Aron. In addition, if you are playing Ruby, you can find Mawile, or in Sapphire and Emerald, you can find Sableye.
Go north, then east. You will pass by a slope to the north, but you can't climb it until you come back with new equipment. Go south from that slope to find a Poké Ball. Go north again, then east, then follow the path southeast to find another ladder. Climb down.
Granite Cave B2F
In this area, you can find Zubat, Abra, and Aron. In addition, if you are playing Ruby, you can find Mawile, or in Sapphire and Emerald, you can find Sableye.
Go north and west from the ladder. Continue going west until you find some stairs leading to the ledge. Go up, then go east. Look for some stairs to the north. Go up those stairs and check on the rock to find an Everstone. Then go down the stairs and continue going east. Go up the ladder to the east.
Granite Cave B1F, part 2
Go directly west and go up the ladder there.
Granite Cave 1F, part 2
Go west as far as you can go, then go southwest. You find a door in the wall. Go through.
Steven's Room
In Steven's Room, you can encounter Zubat, Abra, Makuhita, and Aron.
Talk to Steven to give him the letter. He gives you TM47 Steel Wing. Then he will leave.
Use an Escape Rope, or exit the room and jump down the ledges to the right, and exit Granite Cave.
Go back to Dewford Town and talk to Mr. Briney. Now he will let you go to Slateport City.