Though a saxophone can appear to be a very forgiving instrument, many people don’t expect their first attempts at a note to produce a squeak or faint blowing sound. Rather than trying to blow as hard as possible, try to blow as if you’re trying to fog up a mirror. Hold your jaw in a very loose position and don’t squeeze the reed between your lips to try to control the vibration. It will take some practice to develop a strong enough embouchure to hold the mouthpiece securely in your mouth, and at the same time keep your mouth loose enough to let the reed vibrate and make a clear tone.
A lot of the problems in the beginning come from not using enough air. When it sounds weak, don’t blow harder, that usually just results in tensing up and strangling the reed. Instead, think of the air as being fast, not necessarily strong, more like a jet of water than a gale of wind. Try playing a long note at medium volume, and see if the pitch or tone changes. If it does, start again and aim for a steady flow rather than a strong one. The clarinet responds to this far better than to strength.
One of the most frequent errors is biting to prevent squeaks. It might eliminate the squeaks for the moment, but it kills the tone and causes the upper register to become undependable. To overcome this, use the lower lip as a pillow rather than a vice. The teeth should be gently placed upon the mouthpiece and the tone supported by breath, not pressure. As the embouchure releases its pressure, the reed will vibrate more freely, resulting in a bigger, easier tone. If squeaking is still a problem, be certain that the angle of the mouthpiece is not too flat. That, too, contributes to biting.
An everyday short exercise will give you fast results without any tiredness. Take a few seconds to put your instrument together, then play a note you feel comfortable with for as long as you can, and make sure that the sound is consistent from the beginning till the end. Take a few seconds rest and try it on a neighboring note. Notice that your shoulders are relaxed, your throat is wide-open and you breathe all the time. Practicing like this for 10-15 minutes will do more good for your sound, than just playing around some songs, because it strengthens the base that every other exercise is built upon.
If you hit a plateau, play some long tones and record them, then take a break and come back to them. Sometimes pitch that seems steady playing the instrument sounds slightly wavering when you step away from it. Change only one variable at a time; if you are trying to correct the pitch, change either the rate of your breathing, the pressure of your embouchure, or the position of the mouthpiece, but not more than one of those at a time. This will help you to isolate what effect each has on the pitch. Clean tone is not a magical gift that a select few possess; it is a matter of consistent vibration supported by free tension, and both can be cultivated through diligent practice.