Say I have a graph that has a set of points on it. The points represent some data.
When should I connect the points with a straight line, when should I connect the points with a curved approximation, and when should I not connect the points at all?
Does, in the end, choosing whether to connect the points and how to connect the points come down to some personal belief based on whatever feels reasonable in a certain situation?
Or perhaps there are some "official" rules to follow that must not be violated, and violating such rules would disqualify the graph from being correct?
Connecting points (or not) on the graph is mostly dependent from the codomain.
If your function's codomain is continuous like $\mathbb{R}$ (e.g. in a function $f(x) = x$), then you should connect all of the points on the graph, as the function is continuous (which means that there are basically no gaps between the points), but if your codomain is not continuous like $\mathbb{N}$, then you should not connect them (e.g. in $f(x)=\lfloor x \rfloor$), as the codomain is $Y=\{\dots, -2, -1, 0, 1, 2, \dots\}$ and between every single point there is a gap.
When it comes to connecting points with various shapes (straight lines or curves) it depends from the function itself. Working with linear functions will make you connect all the points with a straight line, but with polynomial, exponential, logarithmic etc. they should be curved, as it is defined by their codomains.