Paddling your surfboard sounds so simple. However, in surfing situations, you are also balancing your body on moving water while getting knocked about by breaking waves. So remember to start small and go out when there are little to no waves coming in.
Your first step in learning how to paddle your surfboard is to walk your board out until you are in waist-to-chest deep water. Lay your body on the deck of your board, being careful to keep your weight centered on the middle of the surfboard. The natural tendency is to lean back towards the tail of the board, but this causes the nose to rise and creates resistance when you're trying to get moving.
Instead, keep your chest just above the center point of the board. Once you feel stable, lean forward and let your nose and tail level out. Now you're ready to get moving! Alternately paddle your arms with cupped hands. The more resistance you feel, the faster you'll go. As you hit bumpy water or "chop", lift your chest slightly and lessen your weight on the board so the nose and rails don't go under.
Remember that you will be balancing your right and left sides, head, and legs all while you paddle your surfboard out to the lineup.
Next up: Learn how to duck-dive.
If you are unsure about references to the parts of a surfboard, read on.