How do I write standard N, R, Z, Q symbols for number sets?

You can write the symbol of a number set (with double lines) using the mathbb font which is supported by amsfonts or amssymb package.

In order to write the notation of a set of real numbers, you can thus use

\mathbb{R}

in math mode if you have included the correct package, i.e. if you either have

\usepackage{amsfonts}

or

\usepackage{amssymb}

in the preamble of your document, so you can write stuff like the following:

$\mathbb{N}$ is the set of natural numbers,
$\mathbb{Z}$ is the set of integers,
$\mathbb{Q}$ is the set of rational numbers and
$\mathbb{R}$ is the set of real numbers.