Popping up is the most difficult thing about surfing when you are first learning.
Surfing is one of those sports that other people make look easy, but when you are trying to do it's strange and awkward and you can't figure out why your body wont work. You might be really flexible, strong and athletic, but for some reason you're struggling with surfing. Most likely the paddling technique and the pop-up are what is keeping you from being the shredder that you imagine yourself to be in your own head.
So, here are some basic tips to get you started.
Chest up
Keep your chest up high while you are paddling. Your chest and most of your ribs should be lifted off the board while paddling around. Think of your torso as a platform connected to the board; the platform edges are at your lower ribs and your hips. Arch your back and lift your chest up. You will have to exercise your back muscles to be able to paddle for long periods of time as this is not a natural position.
Arms bent slightly and stroke close to the board edge
Paddling a surfboard and swimming are two completely different things. So don't try swimming freestyle with your arm circles on top of the board. Its inefficient and you will need to have all the energy you can saved up for charging for your wave. Keep your paddling strokes close to the board edge. This will maximize the amount of forward power you get from each stroke. If you reach your arm out further from the board it tends to propel the board laterally which is counter productive.
THE POP UP
The moment when you are in front of a wave as it builds. You are paddling with all your might to catch it. Then you feel the wave take the board and all you have to do is stand up and ride. If you mis-step or wobble with your pop up then the ride is immediately over before it's begun.
There are two basic stages to the pop up; The moment before the waves takes over forward momentum and you are still paddling to propel the board forward, and the moment the waves takes over. So lets take a look at technique during both of these times.
Before the wave
It's all about timing and placement relative to the wave as it is breaking. The only way to really get a handle on this is spending time on the water. Each wave is different and you have to judge each one separately. To manage controlling your position on the wave as you are paddling for it there are 2 simple techniques. Lifting your chest and lifting your legs.
You have to understand that weight distribution on the board is what causes you to "drop in". Basically, if you are behind the wave you need more weight forward, if you are too far in front of the wave you need more weight back.
Explained in terms of body mechanics if you are behind the wave you would bend your knees and put your chest down, If you are too far in front of the wave you would lift your chest and keep your legs straight.
So imagine you are paddling for the wave, it is building quickly and you find yourself just about to miss it but you just need that little extra UUUMMMPPff to get into it. Obviously, PADDLE HARD! But also, use physics to your advantage and redistribute more weight forward by bending your knees and even arching your back to rock your body forward.
When the waves takes you
Once you feel the wave take control and start pulling the board with out you paddling, that is when it is time to pop up.
Think of your arms as a means to hinge your body to the board. Place your arms on the board edges as though you were going to do a push up then swing your legs under your arms as the board drops in.
When you drop into a wave the board literally drops below you. Use those mechanics to your advantage and stand up when the board drops down.
I hope this helps. Happy Surfing