Let $(\text{Spec}(A),\mathcal{O}_X)$ be an affine scheme. We say that $f\in A$ is a function on $\text{Spec}(A)$ by having for $[\mathfrak{p}]\in\text{Spec}(A)$ taken to $f([\mathfrak{p}])=\overline{f}\in A/\mathfrak{p}$ (meaning $f{\mod \mathfrak{p}})$.
Then we can see that $f:\text{Spec}(A)\to \bigcup_{\mathfrak{p}\in\text{Spec}(A)} A/\mathfrak{p}$, later this is meant to be $f:\text{Spec}(A)\to \bigcup_{\mathfrak{p}\in\text{Spec}(A)} A_{\mathfrak{p}}$ right? So how do I bridge this gap? How do I relate $A/\mathfrak{p}$ to this localisation (in regard to functions and global sections)?
How do I think of a function $f$ on the spectrum as a global section?
$A/\mathfrak{p}$ is a domain, and its fraction field is isomorphic to $A_\mathfrak{p}/\mathfrak{p}A_\mathfrak{p}$, which is also called the residue field at $\mathfrak{p}$. The germ of the element $f$ takes values in $A_\mathfrak{p}$, but when thinking of $f$ as a function, the value of $f(\mathfrak{p})$ is the image of $f$ in the residue field at $\mathfrak{p}$ (or as you said $A/\mathfrak{p}$).
You can also think of $f$ as a section of the 'espace etale' (or something), which is basically like you said where you collect all the stalks together. To get what most people call the value at $\mathfrak{p}$ though, you have to mod out by $\mathfrak{p}A_\mathfrak{p}$.
Given an element $f \in \mathfrak{p}$, - where one would say $f$ vanishes at $\mathfrak{p}$ - note that $f(\mathfrak{p})=0$ in the first interpretation (image of $f$ in $A_\mathfrak{p}/\mathfrak{p}A_\mathfrak{p}$) but often/usually $f$ is nonzero in $A_\mathfrak{p}$.