When to use "a" with "mirar"?

[Christoferus]

I'm a beginner at Spanish, and I'm having trouble with the word "mirar", meaning "to watch" or "to look at" (I presume). I've seen it used both with a the preposition "a" proceeding it and without the preposition "a". My question is when should I use "a" after any form of "mirar"? Here are some examples I've found of "mirar" in use.

Sentences with "a" after "mirar"

They watch their children.
Miran a sus hijos.

He looks left and right.
Él mira a la izquierda y a la derecha.

Sentences without "a" after "mirar"

The cat watches the fish.
El gato mira el pez.

They are watching T.V.
Están mirando televisión.

So is there a rule? Are they interchangeable? Please correct me if any of my information is incorrect. Any further details about the word "mirar" would be much appreciated!