Learning to play the saxophone and progressing with it is all about practice; getting into the right habits at the start and following (at least until you find your own voice) those that have come before; emulating and playing like those saxophonists you admire is a great way to learn.
The saxophone is mainly a jazz and blues instrument and these two disciplines are mostly about self expression and improvisation. But before you can get up there on stage there are many things to learn first.
After buying the saxophone and learning where the fingers go, next will come scales, and there are a lot of them. If all you want to play is blues then the 12 blues scales are a must; but just improvising in a blues scale becomes limiting after a short while. Although it may sound good to you, others who listen will wander off after a while because just playing in the blues scale becomes monotonous.
Knowing the progressions and all the scales and chords means you can be very versatile, so learning the major and minor scales first is best, then the Dorian scales which are used in jazz a lot. Eventually you will know all the scales and practice them every day for at least an hour.
Long notes are important. Doing long notes for 20 minutes or so helps to toughen up the embouchure for strength of sound. Play one note for one long breath and hold the tone firm. Take a breath then play another note and hold the tone steady. You can play all 8 notes of the c major scale doing this and that will be enough for one day, and you should feel it too around your mouth, perhaps enough to stop playing for the day. Don't overdo it; just do what you feel comfortable doing then stop.
Playing long notes and using a different scale every day helps you to learn the scales. When you become proficient at scales you can then put patterns into your practice. There are plenty of books in the music shops to help with learning patterns.
If you listen to the jazz greats playing you will hear them put patterns into their playing.
Always keep your fingers lightly touching the keys, this will help you to play faster. If you get into the habit of lifting your fingers away from the keys as you are playing this will slow you down, and be very hard to put right years later when you get good.
If you find your valves sticking then place a cigarette paper under the leather pad and close for a few moments, this should cure it.
If the read is too hard or is not playing as it should then using a razor or sharp craft knife to gently scrape the part of the read your lip sits on will make it easier to play and sometimes prolong the life of the read by closing up the air holes in it.
When playing for awhile you may find your teeth becoming sensitive after resting on the mouthpiece. This usually happens with metal mouthpieces. The cure is to use a rubber pad bought from a music store that is stuck onto the mouthpiece so that your upper teeth are protected from the vibration of the mouthpiece.
There are many different mouthpieces that alter the sound of your playing. Some make it brighter; some make it sound more earthy.
You of course will have to play as many as you can to find the right one for you. Some of them can be really expensive. The plastic mouthpiece that comes with the saxophone is all you will really need until you become very proficient at playing.
Busking can be daunting the first time you go out alone, but if you have learnt enough pieces of music well enough you will find people will stop, listen and applaud, and give you some money too. Tunnels and underpasses in the town or city where a lot of people pass through are great places to enhance your sound and make it bigger by bouncing off the walls. Picking the right spot can make a big difference.
Playing with someone can make the whole experience fun and enjoyable and gives you a chance to take a break when you need one. Busking is an invaluable experience because it puts you in front of an audience, and this invariably gives you feed-back, some good and some not so wanted. Try not to use too much vibrato in your playing, it becomes stale very quickly, and people will come up to you and say you sound like an old crooner.
When you start getting better you may be asked to do gigs in cafes, clubs, bookshops and etc. Only do this if you feel confident you can play for the full set. But sometimes only experience will let you know that.
When you're busking you have people walking by, but in a cafe etc you will have a captive audience. Big difference.
Practice, practice, practice, is what will get you there, the more the better. You may not become a great player, but with enough practice you will play great.