I recently started reading Mathematical Logic by Joseph R. Shoenfield, and he wrote the following under the section on variables and quantifiers, in the chapter on first order theories:
If the universe is the set of natural numbers, we can assert that every natural number is equal to itself by asserting $x = x$. While we can now assert that every individual has a certain property, we have no formula which means that every individual has this property. To see the disadvantage of this, suppose we assert $x=0$. We would then be asserting, incorrectly, that every natural number is equal to $0$. We might hope to make this into a correct assertion by placing $\lnot$ in front. But to assert $\lnot(x=0)$ is to assert that every natural number is unequal to $0$.
He then goes on to say that by introducing $\forall$, we can assert $\forall x(x=0)$, which means that every natural number is equal to zero. Then by negating it, we get $\lnot \forall x(x=0)$, which correctly asserts that not every natural number is equal to zero.
I understand why asserting $x=x$ is the same as asserting that every natural number is equal to itself, and I can also see why asserting $x=0$ is like asserting every natural number is equal to zero. But I don't understand why asserting $\lnot (x=0)$ is asserting that every number is unequal to zero. The only way that I can see how that would work is if $x=0$ doesn't mean "for all $x$, $x=0$", but instead was like the list "$x=0$, $x=1$, $x=2$, ...", and by negating it, $\lnot (x=0)$ becomes the list "$x\neq 0$, $x \neq 1$, $x \neq 2$, ..."
If that's the case, then what makes the first list different from "$\forall x (x=0)$"?
This is a convention that if you are in a context where you expect a sentence (with no free variables), and you write a formula with one or more free variables, then it's understood to mean the universal closure of that quantifier, where you prefix the formula with $\forall y$ for each free variable $y$.
So if you assert the "sentence" $\lnot(x = 0)$, then since $x$ is a free variable, that is to be understood as asserting $\forall x (\lnot(x=0))$, which you can read as "every $x$ is unequal to 0". (And of course this assertion is false, as witnessed by $x=0$.)
There is no way to write the sentence "there exists an $x$ which equals 0" as the universal closure of a formula without quantifiers, so this motivates the need for both $\forall$ and $\exists$, and the ability to apply further logical connectives to formulas with quantifiers.