Steps

Get into the water, making sure that it is not too deep for you. Holding on to the side of the pool, or someone you trust, let your head go under water. Remember to hold your breath. If you blow bubbles out of your nose then water will not go up your nose.
Grab onto the side of the pool and let your legs float out behind you. This will be easier if you put your face in the water. Start kicking your feet and get a feel for how it's done. When you're ready, stop kicking and proceed to step #3.
Turn around so you are facing the open water.
Slightly jump off the ground and bring your right arm in front of you, cup your hand and push your arm through the water until it's at your side. Repeat the process with your other arm. You are pushing yourself through the water with your arms.
At the same time, kick with your feet, like you did in step #2. This will make you move faster.
If your arms get tired, do not panic! You can flip over onto your back, let your arms float out to your sides, like a 'T', and let your feet float out in front of you. Relax, then just kick with your feet to keep moving, or just kick your legs.

Tips

Make sure there is another person present that is able to swim in case of emergency.
You can also take swimming lessons at most public pools.
If your legs sink while floating on your back, try to put your arms over your head as if you were reaching for something, this will make your legs float up upwards.
Keep your head down in the water. If you lift it up to breathe, your body sinks and stops all forward motion. To breathe roll to your side, head stays down but finds air with your chin just over the high shoulder.
Lengthen your pull through the water. Reach way out ahead, pull straight through centered under your body, then keep pushing down to your thigh before the hand leaves the water. The recovering hand moves in a straight line forward just above the water to reach way ahead again. Think long stroke.

Don't panic! If you get exhausted, flip over onto your back and float - as described in Step #6.
Until you are comfortable in the water do not go into water that will be deeper than your chest. That way, if you get scared or tired, you can put your feet down without too much of a struggle.
If you are really skinny, you will not be able to float, because fat is what keeps you buoyant. You may want to try wearing a life jacket.
When you first learn to swim, you are using your muscles in new ways, and this is extremely tiring. Always have someone nearby who can swim well.
Underwater breathing contests are very dangerous. Although you might think, "we are only a couple feet in the water," if you stay underwater for too long, you can drown quite easily. But don't be scared, just say politely, "No thanks," if someone asks you to have an underwater breathing contest.