Pokémon FireRed and LeafGreen/Four Island

One Island
In order to truly finish the game, you'll need to explore all of the Sevii Islands. To begin, you will have to have defeated and obtained the National Dex from Oak after having caught at least 60 Pokémon. Once those tasks have been accomplished, go talk to Celio, Bill's friend on One Island. To get there, take the Ferry from Vermilion City; Celio is inside the Pokémon Center. Celio asks you to help him find a gem that will make his Pokémon trading machine work. Go to Mt. Ember to find the gem.

In search of the gem
Once you have arrived at Mt. Ember, go right after going up the stairs. You'll find two Team Rocket members talking to each other about the secret passwords ("Goldeen need log"), standing in front of a cave entrance. Go up to them and battle both of them. They'll leave after you defeat them. Go inside the cave and follow the path. You'll go through a boulder puzzle, and then reach a fork in the road. Take the south road; the other road takes you out. The next room is full of Braille messages. Go on to the next room. The Braille message says "everything has meaning existence has meaning being alive has meaning have dreams use power." The Ruby is located in the center of the room below the room with the Braille messages. Leave Mt. Ember, and go back to Celio. Hand him the Ruby, and he'll want you to go on another mission. You'll hand him your Tri-Pass, and he'll give you the Rainbow Pass, which gives you access to the rest of the Sevii Islands. Now take the Ferry to Four Island.

Four Island
Once you arrive at Four Island, also known as Floe Island, Gary will first come to talk to you. First, go north a bit along the beach and break the rock. Keep going north for a Star Piece. Return to where you were and go to the house next to the Poké Mart. Talk to the person inside, and he'll apply a Hall of Fame Sticker to your Trainer's Card. In the upper-right hand corner of the island is a house with a Move Tutor inside, which teaches. Next to that house is the Day-Care Center. Leave two Pokémon here, a male and a female, and they may breed, as long as they're in the right Egg Group. You also can put Pokémon here to raise their levels. Next, Surf across the pond for a Stardust, and then Surf to the other side to enter the Icefall Cave.

Icefall Cave
Once you're in Icefall Cave, surf across the water and go deeper. In this room, you'll find a puzzle with many blue ice tiles. They are just thin pieces of ice that crack the first time you step on them, and break the second time. This is a fairly easy puzzle. If you get to the bottom-right corner of this room, you'll find HM07 - Waterfall. You may need to fall through certain holes to get to the HM, and after some tries you'll get it.

After getting the HM, teach it to a Pokémon, exit this room and go up the waterfall. Go up the ladder, into the entrance, and you'll find Lorelei defending the Pokémon of this cave from Team Rocket. She'll then drag you into the argument, and you'll have to fight a Rocket member. After the battle, the Rocket member tells you that they've kept the stolen Pokémon in the Rocket Warehouse in Five Island. The Rocket Warehouse requires two passwords, and you only have one from eavesdropping on two previous Rockets. Instead of going to Five Island now, you'll need to first go to Six Island to find out the other password.