I am a reasonably competent surfer, but new enough to the sport that I remember what it was like to be a complete beginner. Here are some tips I wish I had been given when I was just getting started:
1) Start with a longboard. You'll want a 9-foot or even a 10-foot board for learning. If you're very small, you might get away with an 8-footer, but whatever you do, don't get a 6'1" potato chip just because all your friends are shortboarders or because the guy behind the counter at the surf shop thinks longboards are uncool. You'll catch more waves, enjoy yourself more and learn much more quickly on a longboard. Even a second-hand longboard will likely be expensive, but that just means you'll almost certainly be able to sell it for the same price if you decide to switch to a shorter board (despite what some people think, longboards aren't "training wheels" for waves--some of the most skillful surfers I have ever seen were riding longboards).
2) Pop-up method. A lot of people will tell you should "pop up in one smooth motion". This is by far the least useful piece of advice given to beginner surfers, and you should ignore it completely. Break your pop-up down into three distinct steps:
a) Place your hands on the board in front of you and do a push up. All of your weight is now supported by your hands and by the tips of your toes, just as in a regular push up. Rather than keeping your back straight, however, you should arch it so that only your upper body is elevated; your pelvis should remain more or less planted on the board.
b) Keeping your hands firmly on the surfboard, bring your knees up to your chest, so that your feet swing in under your body like a pendulum. You want your feet to end up pointing somewhat across the board, with your left foot in front and your right foot behind (reverse this if you are "goofy-footed").
The key point to keep in mind in this step is that your feet move forward to be under your body, rather than your body moving back to be over your feet. A rearward shift of weight on a surfboard puts the brakes on, allowing the wave to overtake you. If you find yourself kneeling on your surfboard at any point during your popup, you are probably making this mistake, rocking your hips back to be over your knees, causing your surfboard to slow down. Avoid this mistake by remembering these important words of wisdom: "bum low, head high".
If you find yourself making an A-frame of your body, legs straight, bent at the waist, with both your hands and feet on the board, it is because you have straightened your legs before you should have. Keep your knees bent until your feet are under your body.
Note that this step SHOULD be performed in one smooth motion.
c) You should now be in a deep crouch, with your hands still planted on your board. Stand up (this step is optional).
So why do people say you should pop up "in one smooth motion"? Because that's how it feels when you put it all together on a steep wave. Watch people surfing for a while, though; you'll notice that on gentler waves--the kind you should be learning on--surfers will often hold the arched-back push-up position for a second or two to be sure they have caught the wave before completing their pop-ups. Evidently, breaking the popup down into discrete steps is not just a mental exercise for learning purposes, but an actual, functional technique as well.
3) Paddle smarter, not harder. The second most useless piece of advice given to beginners is "paddle harder!" Sure, you have to paddle hard to get a wave, especially when you lack the judgement and manoeuverability needed to be at the wave's sweet spot at just the right time, but come on--you're already paddling harder than any of the experienced surfers out there. What you need to do is paddle the right way, using your body posture to adjust your fore-and-aft weight distribution at certain, critical moments.
When you see a wave coming, start paddling to get up a bit of speed. Just before the wave gets to you, keep paddling steadily, but adjust your weight distribution backwards by arching your back, moving your head and shoulders as far up and back as you can (watch some shortboarders sometime to see how they do it). Just as the wave catches up to you, and you feel your speed starting to increase, throw your body down onto the board, simultaneously reaching one arm forward for a final, extremely vigorous stroke (once you have caught the wave, you may have to arch your back again to avoid digging the nose of your board into the bottom of the wave).
I find that throwing my weight forward in this manner is far more effective than simply paddling harder. This explains how an experienced surfer can sometimes catch a wave with a single stroke, while a furiously windmilling beginner just falls off the back of the same wave.
4) Don't grab the rails when popping up. I already mentioned this, but it is worth repeating here as a separate tip: when popping up, place your hands flat on the board in front of you, rather than grabbing the rails. There is a tendency for beginners to cling to their boards like grim death rather than popping up when they slide down the face of a wave, and having a firm grip on the board will make this all the more likely. Later, when you are a good surfer, grabbing the rails may be useful in some circumstances, but it can hold you back when you're learning.
The final two points are safety tips:
5) Fall flat. Just as you should always assume a gun you are handling is loaded, assume that the water below you is shallow and full of sharp rocks or coral. Head-first falls should definitely be avoided.
6) Always be ready to turn turtle. "Turning turtle"--grabbing the
sides of your surfboard and capsizing it--is useful for getting under
waves. It is also useful for avoiding being hit by another surfer who
is getting too close. Don't wait to find out if you are going to be
hit; turn turtle right away if you are in any doubt at all.