Basically, if you learn to play any one of the saxophones, you will be able to play all of them. The fingering is more-or-less the same throughout the entire family. The big divider is size. It is usual for the beginner to learn the Alto first.
Sometimes - (affectionately and not-so-affectionally) -
referred to as the "Misery Stick". This, because of the difficulty in gaining a complete mastery of the instrument. The clarinet was my first instrument and I was a very long time learning it to the nth degree. But it is a beautiful thing to play - when you can play it!
In my musically-formative years the flute was a purely "classical" instrument. Then an American jazz musician started to play jazz on it and its popularity in improvisation soared. These days I cannot think of a top-line saxophonist who does not also have a flute on his/her instrument stand. I am also hard-put to come up with a jazz musician who does not also play - and love
- classical music.
If you are headed for improvisation you will also be well-placed to learn basic keyboard skills. It should be remembered, however, that there is a huge gap between knowing
how to play something, and actually being able to do it! Nevertheless, it is very rewarding to be able to pick a few tunes out of the piano.