After you become Champion, your rival tells you that a strong pokémon was seen going into Cerulean Cave, northwest of Cerulean City.
Reaching the Cave
If you have one of the three flying pokémon (Charizard, Aerodactyl, or Dragonite), you can fly west from the Cerulean City Pokémon Center to reach Cerulean Cave. If you don't have a flying pokémon yet, you can get Old Amber from the side office of the Pewter Museum, and take it to the Pokémon Lab on Cinnabar Island to turn it into Aerodactyl.
Outside the Cave
Outside of Cerulean Cave, you can battle Coach Trainer Harjit, who has a level 52 Rhydon, a level 52 Rapidash, a level 52 Seaking, a level 52 Dewgong, and a level 52 Nidoking. You get 5200 Pokédollars for winning. Afterward, you get TM60 Megahorn.
Cerulean Cave 1F
In the first room of the cave, go north and surf in the water. From there, go north and get out of the water in the northeast corner of the area to find a PP Max.
Get back into the water and go west. When you reach the end of the water, go north, then west to find a Max Repel.
Go south from there, then east, to find a Full Heal.
Then keep going east, and follow the path, to find a Max Revive.
Now go up any of the ladders in this area.
Cerulean Cave 2F
The second floor of Cerulean Cave is a wide-open area with several items that you can find. To the east, you can find a Full Restore and a Max Revive. Southwest from there, you can find a set of 5 Ultra Balls. North of that, there is a Max Lure. Southwest of that, there is a Rare Candy.
Be sure to check the glowing spots on the ground to find rare hidden items. New items appear in those spots from time to time. There is even a small chance of finding Master Balls in those glowing spots. Be sure to have a pokémon following you in this area so that it can look for hidden items as well.
When you are ready to move on, go to the northwest corner of this area and go down the ladder there. After that, go down the ladder to the south.
Cerulean Cave B1F
Go east and go down the ramp, then keep going east. In the northeast corner of the area, there is an Escape Rope.
Go south from there until you can go west, then north. To the north, you will find a Full Restore.
Go west from there, then south down the ramp, then east, then south. Go west to find a Max Elixir.
Then go north up the ramp, then go west, then south, and get in the water. Go west and get out of the water. Mewtwo is to the north! Mewtwo is level 70, so be sure to have strong pokémon in your party. If you have a Chansey, try using a TM to teach it Toxic. Chansey has a ton of HP and should be able to survive Mewtwo's strong attacks (depending on Chansey's level) so it can use Toxic on Mewtwo and defeat it that way.
After the battle, you can use Ultra Balls and Razz Berries (but make sure to wait for Mewtwo to drop its psychic field before throwing a berry) or you can just use the Master Ball that you got from the Silph Co. president.
After you catch Mewtwo, leave the cave. There will be a cutscene and you will get 3 Escape Ropes. The next time you visit the cave, you can go to where you caught Mewtwo, and there will be a Pokémon Trainer there.
Pokémon Trainer Green
Pokémon Trainer Green has a level 66 Clefable, a level 66 Gengar, a level 66 Kangaskhan, a level 66 Victreebell, a level 66 Ninetales, and a level 68 Blastoise that can Mega Evolve. You get 19040 Pokédollars for winning.
Afterward, you get a Mewtwonite X and Mewtwonite Y. After that, there will be a cutscene. After the cutscene, you can take the five Poké Balls that are on the ground.