One of the first things that people notice when you play is your tone. It is important to have a good tone. The saxophone is a wind instrument, so knowing how to breathe properly is very important. You have to use your diaphragm. The best way I know to demonstrate how to breathe with the diaphragm is simple. When you go to bed, lie on your back and watch yourself breathe. Your gut rises, nothing else. That is what you want to do when you are standing up, use your gut to pull the air in. I have heard trumpet players describe it as breathing with your knees. Using the diaphragm ensures that you fill every space of your lungs with air. The next time you see a pro singer or wind player, notice how they breathe. Better yet, notice how their shoulders do not move. Having good air support is vital when playing the sax.
Lie on your back and watch yourself breathe.