Early Game method (before Fly and Misty):
Things you need: A Pokemon with Teleport (Abra), Healing at the Cerulean Pokemon Center, Poke Balls bought before doing the glitch, Having a Pokemon that can handle a level 15-ish Water Type in front, Saving before any of this happens
1. You must find a trainer who's visual range (a ! appears over their head and they walk towards you to fight you) is long enough that it will occur when their sprite is loaded onto the screen. For this method, choose the Jr. Trainer who is hiding in the Grass next to Nugget Bridge on Route 24.
2. Moving south from the top of the screen, press Start just as his sprite loads, go to the Pokemon menu, and use Abra's Teleport (or in later methods, Fly). This will take you back to the Cerulean Pokemon Center. The ! animation will play, but the trainer will not move towards you and you will Teleport. (To practice pressing start as he loads, move against the cliff wall instead of in his line of sight and move south. If you do mess up and get into a battle with him, reset the game to your last save).
3. Once you have escaped the battle with Teleport, you will notice the start button doesn't work. This is normal. Now you must find one of two trainers: The Swimmer at the front of Misty's Gym if you haven't beaten her yet, or the Youngster on Route 25 who is standing with his back against some shrubs and facing north. Whatever you do, DO NOT just talk to or get right in front of the trainer you choose, you have to let them walk to you. Fight them and win. Your Start Menu now works.
4. This step depends on the trainer you fought. If it was the Swimmer, leave Misty's Gym and walk up to Nugget Bridge until your start menu opens on its own. If it was the Youngster, either walk back to Nugget Bridge or Teleport back to Cerulean and walk up to Nugget Bridge and your Start Menu will open. Before closing the menu, get your Pokemon-catching Pokemon out in front.
5. Closing the menu starts a wild battle with Mew.
Note: Do not fight any wild Pokemon or buy anything or heal at the center after fighting the second trainer. It can cause freezes or glitch trainers to appear instead of Mew.
Here's a video of it working