Yes in theory we should all learn everything there is to know and be fluent in everything. 15? there's 12 major and 12 minor and then there's modes and.....I've been playing music in one form or another for 50 years and I'm not fluent in all of it.
In practice key of C and Am will do just fine. When you feel you've mastered that move on to 1# and 1 b. Keep adding a sharp and a flat and if you live long enough....meanwhile you can have a bit of fun in C
You only need one key to play by ear. The thing with reading everything is, it puts another interface between fingers and sound. Start with very simple tunes and add very simple twiddles. Take one tune you like, learn it with the dots, then shut your eyes and only open them when you're stuck. Do this often and eventually you'll be able to play it. More you do it easier it gets.
It still takes me a day to make sense of a tune, a week to learn it and who knows to master it. I think it's the old school report syndrome , must do better.
C on the sop/tenor is G on the alto/bari. So change the key sig to one with another sharp or minus one flat. C goes to G, Bb goes to F etc Same with all the notes. Just pencil them in on the page. Do the song you like best first then when you get fed up of playing it do another one.
Never mind Giant steps, little steps will do.
Colin, as always, gives very sound advice here. Beog is beog, tamm ha tamm, poco a poco, little by little - sound advice, whatever language.
If you really want to frighten yourself look up Jamie Aebersold's free online book (the link is on this site somewhere) and you will find something like thirty (if my memory serves me right) "common (!?!) jazz scales". Stick that into 12 keys and your are up to 360...
BUT - before you fill your saxophone with cement, tighten the strap securely around your neck and look for a suitable bridge with deep water under it (tho', except for the cement, Sonny Rollins did that for four years) DINNAE PANIC!
By all means look up Aebersold and others - but don't think you need to know everything before you can improvise. Download, stow it away, refer to it when you need to see how another scale goes. But don't start learning scale after scale without spending regular practice time noodling, playing with riffs, doing little variations on a melody, exploring any and all the noises and sounds you can get out of your horn AND BEING MELODIC. Memorise melodies. Get good at playing rhythymically and let the notes look after themselves. Gradually build your repertoire of scales, work towards getting familiar with the circle of fourths (fifths if you want to go around the other way) and being able to play all major scales around the circle. Then minor... then the blues scales and bebop scales... but make this only PART of your practice, and whatever you do don't try to "understand" everything. Theory schmeory - it's not really theory at all, but custom and practice, wot people usually do...wot people have found works (probably by accident!).
Above all - HAVE FUN. make it ENJOYABLE. It's the big secret to learning anything.
There are any amount of ideas on improvisation on the web, many quite subjective. No harm at all in reading them - there are many different ways of looking at it, if it strikes a chord (!) and is helpful, great. If it's too confusing, leave it. You may find it helps later on. Some sites seem to assume that you have already got a PhD in music...
A lot of the time when I improvise I haven't the faintest idea what scale I'm using or what key, chord or anything else I've wandered into. It's all going too fast to work it out (even when not playing fast!) and what the hell. I'm probably playing bits of some of the 360 odd scales I never got round to actually learning... plus some Indian ragas he doesn't even mention (now there is a field to explore!)
Look at Pete's stuff on his Taming the Saxophone site - it is very accessible, very sound, and you don't really need to look any further to get you improvising with confidence.
And print off Colin's advice above and pin it to your wall...