There is probably no short answer. So let me step over it:
Natural numbers $\Bbb N$
I personally prefer defining the natural numbers starting with $0$. Your axioms for the natural numbers are essentially the same when doing so, but the further will get easier. The other natural numbers besides $0$ are usually defines via the repeated use of $S$ on $0$, like this:
$$1:=S(0),\quad 2:=S(S(0)),\quad 3:=S(S(S(0))),\quad ...$$
The Integers $\Bbb Z$
For the integers, you consider pairs $(a,b)$ of natural numbers $a,b\in \Bbb N$. You define the equivalence relation $(a,b)\sim(a',b')$ for $a+b'=b+a'$. An equivalence class $[(a,b)]$ of such pairs is an integer. For example
$$n:=[(n,0)],\qquad -n:=[(0,n)],\qquad\text{for }n\in\Bbb N.$$
The Rational numbers $\Bbb Q$
The rationals are also defined via an equivalence class on pairs. This time you take the pair $(a,b),a\in\Bbb Z,b\in\Bbb N\setminus\{0\}$ and the relation $(a,b)\sim(a',b')$ iff $ab'=a'b$. You than have
$$\frac ab:=[(a,b)].$$
The Real numbers $\Bbb R$
The reals can be defined via equivalence classes of Cauchy sequences of rational numbers. Two such sequences $x_n$ and $y_n$ are equivalent iff $x_n-y_n\to 0$ for $n\to\infty$. A real number is then an equivalence class $[x_n]$. Any rational number $q$ corresponds to the constant sequence $q,q,q,q,...$. Another example: an irrational number cannot be represented as a constant sequence of rationals, so you have to choose a convergent one like this
$$\pi:=[3,\;3.1,\;3.14,\;3.141,\;3.1415,\;...].$$
And we have not even talked about how to define the usual operations $+,-,\times,\div$ on these sets. This is the real hard work.
But for more details you should really consider reading a book on the topic because there is never enough space in an Math.SE answer box to explain it here in all details.
