poner
Poner is 'to put' — but it also sets tables, turns things on, gives names, and becomes ponerse for a whole world of change-of-state expressions.
poner (put/place)
Physical placing and abstract putting — and turning on devices.
Pon la mesa.
Set the table.
Poner la mesa = to set the table. Not placing the table — a fixed phrase.
Pon la tele / la música.
Put the TV on / put some music on.
Turning on appliances uses poner, not encender in everyday speech — though both work.
¿Qué pone ahí?
What does it say there?
Poner = to say/read (on a sign, label, screen). Surprisingly common.
ponerse
Reflexive ponerse = to put on (clothing) OR to become (a change of state).
Ponte el abrigo.
Put your coat on.
Wearing clothes: ponerse for the action of putting on. Llevar for the state of wearing.
Se puso rojo.
He went red / He blushed.
Ponerse + adjective = to become (involuntary physical or emotional change). Se puso triste, furioso, nervioso.
Se pone muy pesado cuando bebe.
He gets really annoying when he drinks.
Ponerse for becoming a quality — temporary, change-of-state. Not ser (identity) or estar (current state).
Tricky bit
Ponerse vs volverse vs hacerse vs quedarse — all mean 'to become' but for different types of change. Ponerse = involuntary emotional/physical change. Volverse = deeper personality change. Hacerse = deliberate transformation. Quedarse = ending up in a state. This quartet is genuinely one of the hardest things in Spanish.