Sangem Town
Travel to Sangem Town and speak to a non-player character (NPC) there named Lucas. It is a good idea to have a high-level Pokemon before you make the trip. Mew is hard to capture and you won't be able to catch him at a low level.
PokeRadar
After you have spoken to Lucas, get a PokeRadar in Sangem Town before you travel to Mew's location. Stock up on any other items you need from town and head toward Route 224.
Mew's Hiding Place
You will immediately see a grassy area after starting down Route 224. There are several patches of grass; go to the patch that is closest to you. Walk around in that area and you will find Mew. Be careful, as he is very strong in battle. You can capture him if you defeat him.
Mew Action Replay Code
If you have an Action Replay device, the following code will allow you to find Mew anywhere in the wild. Press the "L" and "R" buttons simultaneously after you have entered and activated the code in your Action Replay device to spawn Mew anytime you want. The code is:
"94000130FFFB0000
D5000000C0000097
B21C4D2800000000
B000000400000000
C00000000000000B
D700000000024620
DC00000000000006
D200000000000000"