‘By’ is good. It changes the meaning slightly. ‘Entering by the school gate’ suggests that there are other ways to enter and that you are talking about the one where you go through the gate. You’ll immediately see a library on your left. (Before you get to a turn) You’ll see a library at the first left. (At the first place you can turn left)

You'll see a library on your left right away.

You can say enter the gate. You could also say enter through the gate. And it’s first left, rather than left first!