Your sole means of locomotion rests in the muscles from your elbows to your chest. As a beginner, you can expect your arms to weary quickly. Which is another good reason for starting out by wading hip to chest deep and catching white water : You don't have to paddle. Your paddling strength will grow with practise. It just may feel very odd at the first time you paddle a surfboard. After several outings it'll feel more normal. Push ups and paddling exercise help. And yes, experienced surfers will be able to tell you're a beginner just by the way you paddle.
Paddling to Catch A Wave
When you paddle to catch a wave, you don't have to paddle fast. You're not trying to get up speed so that you can match the wave's velocity. You're just getting some momentum going so it doesn't take so much energy for the wave to launch you forward.
To catch a wave, you want to paddle a few strokes hard to get the board cruising, then take it easy as you look back at the wave to see what it's doing behind you. When the wave picks up the back of your board, threatening to tip the nise under, then you paddle hard, with both arms, two or three times, to keep the nose out. If you feel the nose might still go under, arch your neck and back to aim the nose up as you blast with your arms. It's really a matter of timing and wave position, not speed or arm strength. If the nose stays out of the water, and you feel the board begins to slide on its own, then you've got the wave.
Getting Up
Put you hands on the rails (edges) of the board under your chest and do a mighty push up. You can cheat and bend your back and knees. As you do your push up swing one leg up under you. Then stand up as quickly as possible. The sooner you're up, the sooner you have control of the board. If you go slow, getting to one knee, then the other, things may get tippy and you fall.
Ideally, you get up quickly at the top of a wave, as you and the board drop down the face, then the semi-weightlessness makes it easy to flip up and draw your feet under you. Experienced surfers get up to their feet in one quick snap. It comes with practise. The first time you're on the beach, you may want to practise pushing up and jumping to your feet while still on the sand. It's easier practising on sand than on tippy surfboard in water.
Standing
Let's say you made it up on both feet. You ride a board facing sideways, not facing the nose of your board. One hip should be toward the nose, the other toward the tail. Your feet should be appart centered in the board. You're in a slight crouch for balance. Down the center of your board is usually a thin piece of wood, called a stringer, built in to strengthen the board. Your feet should be on the stringer. Knees slightly bent. Arms out for balance.
Now your job as a rider is to lean forward or backward, even stepping forward or backward, to make the board plane evenly along the water. If you lean just a little bit to the side of the board, the board turns. Because you're facing sideways, you can control the turning of the board with you ankles and by leaning. Your job now is to lean and move on the board to aim it along the wave.
Three Things You Don't Know
One is that your surfboard is faster than the wave. Most times you can get up, slide to the bottom, and surf away before the wave has time to break (and sometimes not).
Your board also has a tail that you don't know about. As a wave gets very steep, and it looks like you're going to just fall down the face, you can lean back on that tail and it will gouge down the wave face, gently landing you and shooting your forward with great velocity.
Also, as you're riding along a wave face, and it walls up (steepens toward vertical), your surfboard will keep planing along just fine, even thought you seem to be standing on a speeding bookshelf. Surfboards are made for near vertical water.