First of all you need to make sure you have a pokemon that knows fly, and you need to make sure you have NOT defeated the Youngster in route 25 with a level 17 Slowpoke or the Gambler on route 8 (I think he has a Poliwag, can't remember).
Anyways, this is the Youngster I'm talking about:
And this is the Gambler:
You
must avoid battling both of them.
Now for the quest to catch Mew!
Go to route 8 and stand in front of the front door of the house shown in the Gambler picture. Now this is the hard part. Save the game. Now take one step down and quickly press start to open the menu
before the Gambler has time to see you. This may take a few tries. Now fly to Cerulean city. The Gambler should see you fly off.
When you get to Cerulean City, go past Nugget bridge to route 25 and find the Youngster in the picture above. Now fight the Youngster but MAKE SURE he has to walk at least one step to get to you. Don't go speaking to him yourself or standing right in front of him, he must move. Bear in mind you have to win the battle. Now fly to Lavender town.
Now you're at Lavender town, head towards route 8. The menu should automatically pop up and when you exit the menu you should encounter a Mew, which is on level 7.