I understand that the order of the quantifiers of a statement determine the truth value of statement. For example,
$$\forall x \in \mathbb{R}, \exists y \in \mathbb{R}\ \text{such that}\ y^2>3x+5$$ $$\exists y \in \mathbb{R}, \forall x \in \mathbb{R}\ \text{such that}\ y^2>3x+5$$
Will I be correct in assuming the first statement is false and the second statement is true?
On
"For any real $x$ ( For some real $y$ ( $y^2 > 3x+5$ ) )" means that if you take any real $x$, "For some real $y$ ( $y^2 > 3x+5$ )" is true for that $x$. And if you are given $x$ already, "For some real $y$ ( $y^2 > 3x+5$ )" means that you can find at least one real $y$ such that "$y^2 > 3x+5$" is true for that $y$ (and of course for that particular $x$ that was already given). Think carefully about the whole thing and you will be able to understand whether the original statement is true or false.
Now do the same with the second statement. It is more important to have a firm understanding of the semantic meaning of the statements than to blindly try all sorts of "proof methods". Once you understand each statement, you must be able to convince yourself whether it is true or false. Then write down as precisely as possible why you know it is true or false, such that you expect to be able to convince someone else. If you are correct and precise, anyone will surely admit that you have proven/demonstrated/shown it.
You have your truth/falsity the wrong way around. It is true that for any real number, you can always find one that is much bigger, which is the first statement. But it is false that there is some real number which is always (much) bigger than the others, which is the second statement. Both of these arise from the fact that in the reals, there is no "biggest number" - you can always find a bigger one, but never find a biggest.