When other saxophonists hear me play, very often the first question they ask is "what's your setup"? While the equipment you use is important (very few saxophones out there can produce the same resonance of tone or consistent tuning of the Selmer Mark VI), it's certainly not the most important part of developing the sound you want. If you listen to recordings of great players who change their equipment, they generally still sound like themselves. On Michael Brecker's album "The Nearness of You: The Ballad Book", he's using a hard rubber mouthpiece instead of his usual metal one, which produces a slightly darker tone, but his sound is still easily recognized as his own.
In my experience, the tone you produce is determined mainly by your embouchure, and the openness of your throat (how much air you're putting through the horn). In terms of embouchure, I like Jerry Bergonzi's approach of the "No-emboucure embouchure", which he demonstrates here:
So in essence, the approach is to try to get the reed to vibrate with as little hindrance from your mouth as possible.
For learning to play with an open throat, the best exercise I can suggest is to begin each practice session playing long tones on the overtone series. If you're not already familiar with this, this means that you finger a low Bb on the horn (with no octave key), and by changing the air flow through the horn, will be able to also produce a Bb in the middle register, F above that, Bb above that, D above that, F above that, and so forth.
How to actually produce the other notes is something that's difficult to describe, and comes only with experimentation. It's easy to get the notes in the overtone series to play by tightening your embouchure (i.e. biting down harder on the reed), but the resulting tone will be thin, and will cause either tiredness, or worse yet -- pain, in your mouth and jaw. When I practice the overtone series, I concentrate on keeping my embouchure at a comfortable pressure, and producing the notes by adjusting my throat. When moving to the next higher overtone, I picture the back of my tongue rising to meet the roof of my mouth, and this works quite well for me. Playing with an open throat also feels similar to yawning, so imagining you are yawning while playing may also help you to learn this concept.
Playing the overtone series is also the basis for playing in the altissmo range of the saxophone, so you can develop both your sound, and expand your range on the instrument with this single exercise!
Once you are able to effortlessly produce a sound on the saxophone, the next step will be to fine tune it based on your tastes, which is the topic for another article.