I don't understand how to use the water basin (temizuya) at shrine entrances
It's for purifying your hands and mouth before worship. Take the ladle in your right hand and rinse your left hand, switch it to your left hand and rinse your right hand, switch back to your right hand and use the water pooled in your left palm to rinse your mouth, then tip the ladle upright so the remaining water runs down the handle to rinse it.
Never put your mouth directly on the ladle — always pour water into your left hand first and sip from there.
When it's busy, or at shrines that simplify the ritual, just rinsing your hands is often considered enough — there's no need to hold up the line behind you trying to do every step.