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Craft an Iron Pickaxe (wood, stone or even gold pickaxes won't work) so you can receive diamonds when you mine Diamond ore. Gather some food. Bring some weapons.
In order to create an Iron Pickaxe, you will need 3 Iron Ingots and 2 sticks. Diamonds are found in the lowest levels of worlds, so the journey will be a long one. You'll want to ensure that you have enough provisions to able to stay and mine for a good amount of time without dying of hunger. You'll be down in the deepest reaches of the Minecraft world, so you're sure to come across some hostile mobs. Make sure you have some good armor and a trusty sword before heading out. Be prepared!