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Step 1
Get familiar with the location of all the notes and different ways of blowing into the sax. Become comfortable with your fingering and the way the saxophone feels in your hands. Good improvisation comes from being so comfortable with your instrument that it is an extension of yourself.
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Step 2
Learn runs of any key imaginable. There are some basic scales for jazz and blues (music that is most conducive to the saxophone and improvisation). Just as in Step 1, you should practice these until you know them in your sleep. If you want to start jumping into open mic nights and strutting your stuff, you must know the steps.
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Step 3
Listen to famous saxophone improvisers. The most prominent saxophone player of the 20th century is John Coltrane. Regardless of whether you are just starting out or have been playing for years, don't try to compare yourself to Coltrane, just try to learn from him. He was a monumental talent and made a big impact on the music world. Don't frustrate yourself.
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Step 4
Buy an improvisational CD to play along with and improvise. This will be a pretty boring CD if you just want to listen to it in the car. It is basically background music for you to play the lead over in lieu of actually playing with a band. It's a good way to get your style down and practice your phrasings. This hasn't always been available to sax players, so take advantage of it.
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Step 5
Test your skills by jamming with some friends or walking into an open jam night at a club. It will be scary at first, but as you start to play and remember what you've learned, your confidence will grow. The more you practice with a live band, the better at improvisation you will become.
