What are the different types/ genre of saxophone music besides Jazz?
Saxophone is most closely associated with the Jazz and Rock genres. There are plenty of sub-genres that fall within the umbrella of Jazz but that is another discussion. A better way to look at the saxophone is to get to know the different types of saxophones and who plays which. Generally the types are: soprano, alto, tenor and baritone. Get to know how these look, sound and who plays each. You are doing pretty good if you can tell an alto from a tenor by ear. Might be hard for a novice but not for an experienced listener. You will really be doing good when you can tell the difference between two alto players by ear (for example Paul Desmond vs Art Pepper). Yes, they both played alto, but each had his own "sound." Ditto for any other tenor players, soprano players, etc. Approach it this way and you will never get bored.
Two guys worth checking out but not previously mentioned in this thread: Dave Liebman and Mark Turner.