Root or zero...which to use when?

The difference is the following: whenever you have a function $f:X\to \mathbb R$ you say that $x^*$ is a zero of $f$ if $f(x^*) = 0$. On the other hand, the same $x^*$ is a root of the equation $f(x) = 0$. The rule of thumbs: zero refers to the function (e.g. polynomial) and root refers to the equation.

In general we will find zero of a function root of an equation

Don't use "root" in Australia. It has a very different meaning.