A function is bijective when following is true
$$ ∀ y \in Y, !\exists x \in X : f(x) = y.$$
How can someone read this notation? I know it should state "All elements from Y have one element from X", but I have a hard time to read the notation (especially the $!\exists x \in X$ part) in a mathematical way as a beginner.
And would this be the same as: $$ !\exists x \in X, ∀ y \in Y : f(x) = y?$$
The exclamation mark indicates "unique".
Your statement is pronounced: "for all $y$ in $Y$, there is a unique $x$ in $X$ with $f(x)$ equal to $y$".
Your other statement is not the same at all. Consider $f$ the identity function (definitely bijective), and ask yourself if the statement that "there is a unique $x$ in $X$ such that for all $y$ in $X$, $x = y$" is true.