Believe it or not, Mew can actually be captured in Pokemon Red, Blue, and Yellow through an easter egg glitch. Previously, people thought it was only obtainable from Nintendo Events. We later learned that anyone can get this Mythical Pokemon by following the steps below:
Do not battle the Youngster Pokemon Trainer on Route 25 (North of Cerulean City). This kid has a Slowpoke. He is the fourth trainer after the “Nugget Bridge” that was staring upward.
Do not battle the Gambler Pokemon Trainer on Route 8 (West of Lavender Town). This guy had a two Poliwags and a Poliwhirl. He was the last trainer to the west that was staring upward towards the Underground Passage house.
Obtain HM02 Fly and teach it to a Pokemon. It’s best to have a lower leveled Pokemon that knows a Status Ailment attack on your team. Make sure you have some Great or Ultra Balls in your bag before attempting the Easter Egg.
Go to Route 8 and stand in front of the Underground Passage’s door. Make sure you save before doing any other steps. Take one step south and immediately pause your game. Use a Pokemon that knows Fly to reach Cerulean City. If you did this right, the Gambler will see you (!), but he won’t be able to challenge you to a battle since you flew away.
Walk north to Route 24 and 25. You will need to battle the Youngster Pokemon Trainer with the Slowpoke that is looking upward. Make sure you are walking up against the North wall so the trainer walks towards you, or else your game will freeze up. Defeat the trainer and fly to Lavender Town.
From Lavender Town, walk West to Route 8. Your pause menu will pop up after a few steps. Close the pause menu and you will immediately be thrown into battle with mew! Mew will be at level 7 with the Pound attack. Drop his HP a bit and give him a Status Ailment (Sleep is best), then catch him as quickly as possible!