Quick answer
salir = to go out, to exit a place, to depart. Also "to go out" socially.
irse = to leave, to go away; the focus is on abandoning where you are, not on where you are heading.
salir de takes the place you exit; irse a takes the place you head for.
Side by side
Memory trick
Is the sentence about the door you go through, or about the fact that you are no longer there? Door means salir. No longer there means irse.
Typical mistakes
Salgo, hasta mañana.
Me voy, hasta mañana.
Announcing your departure to the people around you needs irse. Salgo on its own sounds unfinished and the listener waits for "de dónde".
Me voy de casa a las ocho cada día.
Salgo de casa a las ocho cada día.
For a daily routine of walking out of the house, salir is standard. Irse de casa strongly suggests leaving home for good.
Check yourself
At the end of a dinner you stand up and say: