Instructions
Play the game until you have access to Ecruteak City and defeat the gym leader to get your badge. This enables you to use the "Surf" ability, which allows you to travel across water. Before you can use "Surf" you must acquire the HM (hidden machine) for it .
Go to the house just north of the Pokecentre in town and defeat the Team Rocket grunt that is pestering the girl on stage. A man will approach you after the battle and give you the HM for surf.
Go to Pallet Town in the southwest part of the Kanto region. Walk to the bottom of town and use your "Surf" ability. This gives you access to Route 21, which is mostly water. This route connects to Cinnabar Island.
Go to shore on the northwest side of this route where you see the tall grass. There is only one area in this entire route that has a section with grass, so you can't miss it.
Walk around in the tall grass until you encounter Mr. Mime. You will mostly encounter Tangelo, but every so often a Mr. Mime appears.
Use non-lethal moves such as "False Swipe" to lower Mr. Mime's health until it is in the red, and then throw a Poke Ball at it to catch it.