Assuming you've found a combination of sax, mouthpiece, and reeds that allow you to easily produce different sounds on the sax, here are some exercises that I found useful for developing the tone I wanted.
When I first pick up the sax for the day, I'll generally play some long tones in the lower mid-range of the instrument (middle C down to about G). After that feels comfortable, I play long tones on low Bb, and play up the overtone series (Bb below middle C, F, Bb, D, F, Ab, Bb, C, D). When I do this, I concentrate on keeping my embouchure pressure steady (i.e. not biting down harder on the reed), and getting the next tone by lifting the back of my tongue toward the roof of my mouth.
The purpose of this exercise is to get yourself in the habit of producing these different overtones with the minimum amount of physical effort. I generally don't spend more than 10 minutes or so on this.
For me, playing long tones becomes boring very quickly. The most success I've had in improving my sound has been when I practice by playing along with recordings of sax players I like. The first saxophonist to really grab my attention as a teenager was Dexter Gordon, specifically his album Go. The first solo I ever learned from start to finish was from the tune 'Three O'clock In the Morning'.
The notes and rhythms are pretty easy to learn, so I would loop small sections of the tune (sometimes just a short eighth note phrase), and concentrate only on matching Dexter's tone, articulation, and vibrato as closely as possible.
This is a process of trial and error, but when practicing it, you'll be teaching yourself how to manipulate your embouchure, mouth, and throat in order to change the tone that comes out of the sax. Practicing by emulation also helps to keep practice sessions interesting and productive, and I think you'll find that you see improvement in your sound much more rapidly than by practicing long tones alone.