Quick answer
ir = to go somewhere; the focus is the destination and it normally takes a.
irse = to leave, to go away; the focus is the place you are abandoning.
Me voy on its own is complete. Voy on its own sounds unfinished.
Side by side
Voy a casa. / Me voy a casa.The first states where you are heading; the second stresses that you are leaving here now.
¿Te vas?Are you leaving? — nothing needs to follow, because the point is the departure.
Vamos a comer. / Nos vamos a comer.Let's eat versus we're off to eat — the second announces the exit.
Memory trick
Naming a destination is ir. Announcing a departure is irse, and it works with no destination at all.
Typical mistakes
Voy, hasta luego.
Me voy, hasta luego.
Saying goodbye means announcing a departure, so it needs the pronoun. <i>Voy</i> alone leaves the listener waiting for a destination.
Me voy al médico mañana a las diez.
Voy al médico mañana a las diez.
This is a plain appointment with a destination, so plain ir is enough. The reflexive would stress leaving the house.
Check yourself
Fill in: "Ya es tarde, ___ a casa."