Sing me to Heaven: Everything you ever wanted (or didn't want) to know about the Saxophone

Origins

The saxophone was designed and built by Belgian instrument tinkerer, Adolphe Sax in the early 1840's, and was officially patented in 1846. In it's outline, a saxophone is really a mishmash of different instrumental concepts, combining the brass, conical-bore body with a woodwind's fingering system and a single reed mouthpiece. The result is a very mellow and rich tone and great tonal flexibility, especially in the upper range. Most people will recognize the saxophone from a marching or military band, or from it's adopted genre, Jazz. However, there is very broad repertoire for what is known as the "classical" saxophone.
There are five typically used Saxophones, whose written ranges are all the same. The normal written range of the Saxophone is from B♭ to f3, although I have included the transpositions for the sounding ranges of the various instruments.

Soprano - M2 lower
Alto - M6 lower
Tenor - M9 lower
Baritone - M12 lower
Bass - M15 lower

There are also a sopranino and contra-bass saxophone, but I have never seen or heard those played. Most everyone can learn to use a saxophone no matter what age. It is a hardy and versatile instrument, and much easier for the beginning woodwinds student to learn then the clarinet, because the reed on a saxophone is thicker, and thus vibrates with more back-pressure.

Putting the pieces together

One should generally sit down while putting the saxophone together. Less chance of disaster, which is why the mouthpiece cap can be used to cover the reed after assembly. Must protect the instruments! General assembly is as follows;

Put the mouthpiece on the neck with a gentle sliding or twisting motion, using some cork grease if the cork is too tight. One should be really careful with this step, because the placement of the mouthpiece on the neck affects how in tune the instrument is. Pulling the mouthpiece out will lower the pitch, and pushing in will raise the pitch. A general guideline is to put the mouthpiece about one inch into the neck, and then adjust from there.
Then connect the neck to the saxophone body. You should sit, holding the body in your lap with your left hand around the bell, making sure not to grab around any of the keys or valves, as any damage will affect the tone of the instrument. Make sure that the tightening screw is loose, pick up the neck with your right hand, and align the bottom of the neck over the center of the keys. The octave key valve and rod should be lined up, and then just slide into place. You can use some oil if the going gets rough, and then tighten the screw.
Place the ligature on the mouthpiece, watching to make sure that the right side is up. Slide the thick end of a moistened reed into the ligature and line it up, so that the tip of the reed creates a seal with the facing of the mouthpiece. Tighten the screws, and you are done! You are ready to play, or at least try.

Playing a Saxophone

Breathing.
Really, I can not stress enough how important it is. Not just for life (duh) but for proper playing technique. To optimize your breathing, one needs to learn how to control the diaphragm, the bowl-shaped muscle on the underside of your ribs, which separates the lungs from the stomach, and allows for exhalation and inhalation. The action of the diaphragm is largely involuntary, but as all singers and woodwinds players know, you can learn how to manipulate the diaphragm, with other, voluntary muscles to maximize your breathing capacity. There are lots of exercises you can find to help you with this, the best of which will try to get you to hold your diaphragm expanded, in its flattened shape, for as long as possible to increase breath control.

Posture
Posture is also critical here, to avoid strain and increase breath capacity. The saxophone uses a neck-strap to help balance the weight, but otherwise the weight rests on the thumbs, under the thumb-rests. The head should be held erect, bringing the mouthpiece to your mouth, instead of bending forward to reach it. Your top teeth should feel like there is pressure on them, and very little pressure on the bottom teeth. The back should be straight and relaxed, with both feet firmly planted on the floor.
Your hands should be relaxed and arched, but spread and ready to play. The keys don't require a whole lot of pressure to close firmly, so being relaxed is the most important part of hand positioning, which leads to more efficient fingering.

Embouchure
There are two types of Embouchure, the "classical" and the jazz, but I am only going to talk about the classical one here. The classical embouchure involves a firm, but not tight mouth seal, like a rubber-band, around the mouthpiece. The mouth should remain relatively still during all playing, and the teeth should line up as if there were no instrument in the mouth at all. To form the embouchure, roll the bottom lip over the bottom teeth, place the top teeth about one-half inch from the tip of the mouthpiece and seal the lips. The throat should be really relaxed, as if one were saying the syllable "ahh".

Tonguing
On the saxophone, all articulation begins with the tongue. To begin, assemble the instrument with a thoroughly soaked 2-2 1/2 reed and hold in playing position. Take a breath and form your embouchure, then place the tip of your tongue gently against the reed, pushing it to create a seal. Start the air pressure and lift your tongue off the reed to initiate the tone. When you wish to stop the tone, simple cease blowing air. Steady breath supply is crucial, and one should be careful to avoid puffing the cheeks too much. Most tonguing problems are the result of changing registers, harsh attacks, neglecting to use the tongue to start the tone, and not touching the tongue to the reed completely. This is a helpful checklist to follow if one finds they are having difficulty: