After you enter the cave, go up the stairs and then walk into the water.  Surf over to the next plateau (it's on the left side of the room.)  Get off this plateau by taking the stairs at the bottom of it.  Do not leave this level of the room!  Keep going left until you reach another plateau.  Climb up the stairs and go down the ladder you find there.  You should be taken to the next floor down.  Go up until you reach the stone barricade.  Now go right until you come to the second fork in the path (there should be two paths, one going down, and one going straight a step, then up.)  Take the downward path and follow it until you reach another fork.  Turn left and follow the winding path upwards until you reach another fork in the road.  Take the upper path, and then go left and up again until you reach a ladder which takes you back up to the first floor (but to a part you couldn't reach before.)  Follow the winding path downward until you reach another ladder which will bring you to the lowest floor of Mewtwo's Cave.  Go to the right side of the plateau you are on and go down the stairs.  Take the left hand path and follow it until it leads you to another staircase.  Go up onto this plateau and quickly turn down until you reach the edge of the water.  Surf along the path until you reach another staircase.  Climb it and follow this final land path (it's very striaght foward) until you reach another set of stairs that leads to the water's edge.  Surf from here to the final plateau.  You'll see a creature standing in this middle of that plateau.  It's Mewtwo.  Go up to him and speak with him.  He'll say "Mew" (not much for words, like in the anime) and then attack you.  Hopefully you'll be ready for him.

The best way to capture Mewtwo is with the masterball.  That way you never have to fight him, and he has no chance of escaping.  If you do not have a masterball SAVE BEFORE YOU FIGHT HIM!  He's very tough and powerful so he might kill you before you can capture him.  Also, since he's so strong chances are he'll break out of any ball below master many times before he finally stays put.  When you have him with dozens of stat alignments, and are still trying to weaken him enough that he will quit fighting, you might accidentally kill him.  Saving beforehand allows you to try again until you get it right.  Remember too, that he's a psychic pokemon.  If you want to put him to sleep, you'll have to use sleep powder and not a psychic sleep attack.

Game enhansors. 

Get out your Gameshark and plug in this code 0115D8CF for the US version, or 0115BFCF for the Red and Green Japanese versions.  Now every battle you fight will be a garbled mess of pixels called "mew".  However, after you weaken and capture this pile of junk it will become a normal looking, normal acting safe to use mew.  It will learn attacks normally, level up and do all those other things that pokemon do.  It will have all the ablities of a mew (despite Nintendo's claims that it's not the same.)  and best of all it will not have any of the problems that a traded pokemon has.  It will be loyal to you, it will grow at an unaccellerated level (therefore it won't end up weakenned from accellerated growth.), and you can have another one if you restart the game.  Remember to turn off the Gameshark after you've captured your mew.  I have never had a problem with this code, but I've heard there is a chance it could harm your game.  So far so good with me, but use it at your own risk.

If you have a Game Genie, and you want a mew, don't start bashing your head into a wall for not having bought a Gameshark instead just yet... you can have mew as well!  I have personally, never tried these codes, but I have been told that they allow you to catch mew only on the field west of Cerulean City.  In the Red version, the code is 151-91A-7FC  151-93A-F7E 151-9BA-5DB, only the first line is nessary, but the second and third improve your chances of encountering a mew.  Same deal with the Blue version, except the code is 151-91A-7FC 151-93A-F7E 151-9BA-6EB.  I haven't tried thes codes, so I can't guarantee their safety.

In the Japanese version of Red and Green, it was possible to catch mew another way.  Of couse this way also released a virus into your game which corrupted your save file, and that of anyone you traded with.  This method transformed a koiking (magicarp) into a myuu (mew).  However, the created mew never had the attacks or zukan (pokedex) entries of the real deal.  Due to the destructive and worthless nature of this code, I'm not including it here.

If you don't want to cheat to get a mew, then your only other option is to wait for Gold and Silver.  It is rumored that mew will be a regular (but somewhat rare), pokemon in that game.  This is very believable, since the Japanese Zukan entries for mew show that it is becoming more common.  It was upgraded from "extinct", to "mirage" to "ocassionally sighted" and it said that sightings continue to increase.  It will probably be like a Celfairy-ish rarity by Gold and Silver.

Also, to set the record straight Mew CANNOT BE CAUGHT in ANY OF THE FOLLOWING WAYS: 

•  By not allowing the St. Anne to leave, and then surfing to and moving the truck.
•  By getting all 150 Pokemon and talking to any variety of people.
•  By bringing keys to any person.
•  By taking the amber pipe.
•  Through any sort of random battle caused by anything aside of Game enhansors.
•  Any other way aside of the methods listed above the red letters.