Safety first, if you want to learn to swim (as an adult), make sure you try it in a swimming pool where you can easily stand up if things go wrong and have supervision around in case of an emergency.
As someone who has learned several children to swim, I'd say its pretty hard to learn it yourself properly. Why? Because you can read the words, but that doesn't mean you'll be able to correct yourself if you do something wrong. But if you really put your mind to it, I do think it can be done.
The usual order of learning to swim goes sort of like this:
- Get used to water, lose the fear of getting your head under water, learn how to hold your breath and breathing out under water. As an adult this step should be easy, but don't underestimate it. If you want to swim, you need to accept you're in the water, stop trying to keep as much of your body out of it.
Learn how to float. This is by far the most important thing to learn if you want to swim. You see, your body is perfectly able to float without you having to anything. They key to this is exhaling deeply, filling your belly with air and relax as much as possible. While its easier to float on your back, since you can keep breathing, new swimmers will often let their butt sink down which will hurts your floating.
Start floating on your belly. Basically start at the edge of the pool, pull up your legs and push off. Then try to float in a straight line until your feel the urge to breath and stand up again. Try to have a position like in this image from Swimator Note: you should be floating at the surface, just mimic the posture:
Keep practicing this until you can float like this for at least 10 seconds without 'tumbling' over. An important thing to learn is to put your head into the water, trying to keep your head up, will 'push' your back down and ruin your floating position.
Next move on to floating on your back (watch your head, seriously!). You simply start in a reversed position compared to floating on your back. Hold the edges of the pool while pulling up your legs, then push off. Try to keep the same position as on your belly, putting your head between the arms and keep it in the water.
In contrast to floating on your belly, you should practice to maintain this position for a longer period of time. This requires you to breath in and out of a relaxed and controlled fashion. You'll notice that you start to sink when you breath out or that your buttocks start to sink deeper, so you'll have to learn how to adept to these situations. Have a look at this ehow video for more information
Learn how to move while floating. Once you've really nailed the floating, its time to learn how to move while still trying to float. The reason I tell you to push off the side of the pool is because this keeps you moving. Like a bicycle, its easier to maintain your balance if you're moving than standing still. Likewise, kicking your legs can 'push' you upwards a little bit to make floating easier.
However, remember that you shouldn't have to move to be able to float! This is important, because you'll often see poor swimmers kick like crazy, while not moving very far. That's because the water has a high resistance, if you increase your frontal surface, you're effectively slowing yourself down. That's why all swimming movements are cyclic movements and when you make a stroke, you want to use that energy as efficiently as possible: by keeping your body stretched out and simply floating.
Because it can be quite difficult to learn how to move both your arms and legs at the same time, its often easier to start with the legs first, simply kicking your legs while floating in your back or belly. The first leg movement to learn would be the breaststroke leg movement. As this ehow video shows, you can practice this out of the water. Preferably you have somebody film you or help you guide to learn the motions properly. This may feel silly, but there are countless of 'self-learned' swimmers who have a very inefficient kick, who kick without actually pushing themselves forward.
One last word of advice, don't get overconfident of your swimming skills. Take the time to practice the basic skills first, before moving on to learning strokes. I can guarantee you that better basic skills will do more for your safety while swimming than learning any stroke at all.
Feel free to come back with any followup questions if you have trouble with any of these steps and in the mean time, have a look at several other swimming resources.