Here's how to get mew in pokemon red/blue!!!:

Step 1: Catch a Jigglypuff.
Step 2: Raise it to about level 15 or 16, and evolve it with a moonstone.
Step 3: Get to cerulean city and catch an abra in the route above(Use your Wigglytuff's sing to keep abra from escaping)DO NOT BATTLE THE TRAINER ABOVE THE GRASS!!!
Step 4(optional):teleport back to the pokemon center in cerulean.
Step 5: Make sure the trainer mentioned in step 3 is off the screen by 1 space, be in the trainer's sight, and take 1 step down and quickly press start. The trainer's "!" should not appear yet.
Step 6: teleport away, NOW!!!! The trainer's "!" will appear, but you will still teleport.
Step 7: Your start button shouldn't be working. that's a good thing. now go to the route east of the guy you teleported from, and battle the kid with the slowpoke, and beat him.*
Step 8: Your start button should work again. teleport to the center again and head to the bridge.
Step 9: As you cross the bridge, your start button will open automatically. press B and Mew will appear!!!!
You can use Wigglytuff's sing again to help catch it. I suggest you have about 20 pokeballs and if manageable, a few great balls.

*if you press A on the slowpoke kid, the game will freeze. make sure he walks to you.