You need a pickaxe (any type of pickaxe works). Coal can be found almost anywhere, from inside caverns, cliff faces, and underground.
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You don't.
Coal is very common. Caves are a good source of coal, but if you are mining, coal is most common at y=48.
A single piece of coal can perform 8 furnace operations in minecraft 1.6, so that means 8 pieces of cobblestone.
Underground
You can find coal veins almost anywhere. Look at the side of stone cliff faces.