Precedence in Math (When to use a symbol when it has 2+ meanings?) Equation in question:
\[ \forall n \in \mathbb{N} : n^2 \ge n \]
The symbol $:$ is a symbol for "such that"
$\forall$ for all, for any, for each
When do you know when to use: for all, for any, for each? Does it matter which one you use, or are there rules to follow when selecting one?
I say it's something like:
$n^2 \ge n$ is true for all/for any/for each $n \in \mathbb{N}$ 
But, the $:$ means such that, so then it changes everything, for me. Because then it could be
$n^2 \ge n$ is true for all such that $n \in \mathbb{N}$? I'm so confused.

There are two basic logical "quantifiers" commonly in use. These symbols help specify when a statement is true. Let's take a look at your original statement.
$$∀n∈ℕ:n^2≥n$$
This says "For all natural numbers, $n^2\ge n$". It is not specifying that n is a natural number greater than or equal to $n^2$, it is making a logical statement
$$P(n) = n^2\ge n$$
and then asserts that $P(n)$ is true for each natural number n. In general, 
$$\forall x\in S: Q(x) $$
is a logical statement that says $Q(x)$ holds for all x in S. 
The other logical quantifier is $\exists$, and if $P(x)$ is a logical statement, then 
$$\exists x\in S: P(x) $$
is read "There exists some x in S such that P(x)", and it means that somewhere within S there is an x that fulfills P(x).
