How to Catch Mew in Pokemon Yellow
The journey to mew begins in Cerulean City. After you defeat all trainers on Nugget Bridge, you will move up and to the right, on your way to Bill's house (Route 25). On the way, you will find many trainers waiting to battle. Save your game before you begin to battle, in case you mess up your chance by accident. DO NOT battle the Youngster with the Slowpoke at level 17. He will be the fourth trainer, and standing directly above a female trainer and facing North. Avoid him.
You can now continue on with the game normally until you arrive at Lavender Town. After you pass through lavender, continuing towards your left (Saffron City), you will be passing through Route 8. Again, save your game. Just before you arrive at the entrance to Saffron City on Route 8, there will be a Gambler with a Poliwag and a Poliwhirl. He is facing North toward the door of the Underground Path. DO NOT battle him!
Continue on with the game, progressing to Celadon City. In the house just West of Celadon, above Biker's Lane, you will receive the HM02: Fly. Teach this to a Pokemon of your choice. Once you have taught a Pokemon how to Fly, you must Fly back to Route 8 (between Lavender and Saffron). Stand just one step in front of the underground path - you have not yet battled the Gambler. SAVE THE GAME. Saving after this point will disrupt the process.
Walk down one step, and IMMEDIATELY press Start. The menu should come up before the Gambler "sees" you. Select a Pokemon and fly to Cerulean City. The Gambler will "see" you right before you fly away.
Your Start button isn't going to work from this point on. Go North across Nugget Bridge, and right until you find the Youngster that you avoided earlier. Do not walk up to him, or the game will freeze. You have to let him see you and come to you. Battle him, and make sure you defeat him. Your start button will work again.
Fly to Lavender Town right away. Walk West into Route 8 again. As soon as you enter the path, the menu will pop up on its own. Press B to exit.
There's Mew!! He will be at level 7 and his only attack will be "Pound". Make sure you've got plenty of Pokeballs, he will be tough to catch. Once you catch Mew, you can save with no problem. Enjoy your new Mew!