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What model year is your C6? If it's an 07, look at pages 97 through 99 in the 2007 owner's manual.
Essentially you have to put the transmission selector between the console and the lower dash to "S" in order to use the paddles. You can still drive the car without using the paddles. But if you want to use the paddles, push one of the paddles to shift to the next higher gear, or pull on one of the paddles to shift to a lower gear. If your car has the HUD, the gear you're in will be displayed on the HUD.