I daresay you have already researched this subject online. And maybe you're even more confused than when you started. Or, at the very least, you are no closer to a final decision. The internet can be a confusing place!
My first mouthpiece - (as I remember) - was a 4C equivalent that came with the instrument, and I had convinced myself that I sounded okay. How wrong can you be? A few weeks later, and under some expert guidance, I had a mouthpiece that suited my embouchure, jaw, and lung-power. I was suddenly playing better, sounding better, and making fewer squeaks. My progress improved exponentially, and so did my self-esteem.
I guess what I am saying is: you can own a saxophone that cost the earth, and you can sound terrible. On the other hand you can own a cheap student model and sound like an angel. In saxophone-playing the result (to a large extent) is determined by the mouthpiece/reed combination you actually have in your mouth. Plus, of course, how you hold it there!
But it even goes beyond this. If you have a rotten reed attached to that mouthpiece the sound will be bad however much you spent!
The answer? Take expert guidance - (on-the-spot, for preference) - that you will probably need to pay for!
It is probable that the reed which came with your instrument was a "1" or maybe a "1.5" - the lowest possible strength. This is a shove-it-in-your-mouth-and-breathe strength reed. A vacuum cleaner on reverse thrust would get a sound from this reed, whichever mouthpiece it was attached to. It is possible that your developed embouchure could require such a reed, but it's unlikely. Highly. In my experience 2 or 2.5 is the norm.⫘ What I look for in a reed ⫘
Charlie Parker - (by way of an extreme example) - blew a strength 5. Which is very nearly a floorboard!
For information only...I use "Rico" 2.5(Strength) on Alto. "Vandoren" on clarinet. Hybrid reeds on soprano, tenor and baritone.
Hybrid? - Contact me and I'll tell you.
There are certain keys on the saxophone which operate almost independently. These keys are allowed to open, as opposed to being operated by physical contact of some kind. Whether they work or not is dependent upon the strength of a spring...plus the contact between the pad and the rim of the tone hole. If that contact is sticky with sugary spit that has been allowed to coagulate - is that the word I want? - then the spring itself cannot cope. Result...the key does not open, and you could find yourself playing a G-natural, when all the world and his brother wants to hear a G-sharp! Or, maybe, a B-flat, when the passage requires a B-natural. These things are to be avoided if possible. Stands to reason.
The picture shows the build-up of "stuff" that can gather in the groove of the above pads in particular, right where they settle over the rim. Left is the pad of an "E" key, right is the pad of the "G-sharp" key. Put in the case wet and allowed to dry, it'll harden like glue, holding the pad to the rim of the tone hole!
Before you blow a note, press the G# key to engage the spring, then make sure that the pad has actually opened. If it hasn't, un-stick it. That's every session - practice or performance.
Long-term...Always ensure that the contact between pad and rim is clean (That's before you start playing after any kind of a layoff, where spit has been allowed to dry). A sheet of blotting paper used to be a standard consideration, like making sure your sling was in the case, and not still dangling on your music stand at home! I am not certain that blotting paper is available anymore. Who uses a pen with ink in it these days? Something like newspaper is almost as good. Or tissue of some kind. Any kind.
You may have guessed that the G# key is the main culprit. The B-flat button key can also be a problem, as can the F-sharp key.
Slide your paper - or whatever it is you are using - in between pad and rim, close the key on it gently and, just as gently, ease the "paper" out against the pressure of the pad. This drags out any sticky residue. Though, now that I think about it, you can buy a proprietary gizmo, specially designed to do that job. (The corner of some newspaper or other is way cheaper, of course)
Remember, if your tipple is coke, or some other sugary drink, the problem is clear and present, and it could mean the difference between sounding like a pro, or appearing as if you didn't know your key signatures in the first place. Embarrassing, right?
The question of tuning is, in many cases, totally misunderstood. And it is misunderstood - and therefore mistreated - because it was never explained properly in the formative learning period. Many young students simply take their instument out of its case, put it together, and start blowing. This, of course, is fine if your aspiration is to stand in your front room, on your own, serenading the walls. It is not so fine if there are two or more of you, playing together, whatever instruments are being played. It would not be too far from the mark to say that, with individual tuning ignored, cacophony can insue. And in some cases it is a cacophony that - in the heat of the moment - you may be totally unaware of.
What you find at the end of the following link is arbitrarily cruel, and it would not be available at all if the choice was mine. A natural morbid curiosity probably means that you are going to click it...right? Please remember that it should never have been uploaded in the first place, but it does underscore my point. It is not the kids' fault. It is the fault of the grown-ups in charge of the enterprise. However, since it is up there on YouTube for all the world to see anyway...
I do not intend to delve any deeper into this question here - there are simply too many facets that need to be addressed. Instead, here are a few bullet points you may like to ponder, and maybe bring up with your tutor. Even if you do tune up at the start of a session, your instrument will get sharper as you continue to blow it. Rule Of Thumb: Warm Instrument = sharp. Cold instrument = flat. This is especially true of the saxophone, or any other metal instrument. The metal flute, for example. If your embouchure is not secure you will probably blow flat in the low register, and sharp in the upper register. You rarely sound the way you think you sound. The world beyond the bell of your saxophone is not the world you are living in; but that is the world you are hoping to impress. See #cacophony above. It is helpful if you can be two people; the player and the listener. NB: the more proficient you become, the more you are able to achieve this, simply because you will need to concentrate less on the notes you are playing, and more on the sounds you are making, either as a solo instrument or - more significantly - as part of an ensemble.