Can I ever go wrong if I keep thinking of derivatives as ratios?

$\frac{dy}{dx}$ can indeed be thought of as a ratio. The question is, what do $dx$ and $dy$ represent in this ratio? The answer is, changes in $x$ and $y$ along the tangent line to the curve at the point in question, rather than along the curve itself. See e.g. https://www.encyclopediaofmath.org/index.php/Differential

The chain rule in two dimensions is counter-intuitive at first if you think of the derivative (and partial derivatives) as ratios:

$$ \frac{\mathrm d f}{\mathrm dt} = \frac{\partial f}{\partial x} \frac{\mathrm dx}{\mathrm dt} + \frac{\partial f}{\partial y} \frac{\mathrm dy}{\mathrm dt}. $$

Here, $f$ is a function of two variables, $x$ and $y$, both of which are functions of $t$.

This question has come up at MathOverflow:

https://mathoverflow.net/questions/73492/how-misleading-is-it-to-regard-fracdydx-as-a-fraction

and overlaps with other questions on this site:

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

If $\frac{dy}{dt}dt$ doesn't cancel, then what do you call it?

There are several interesting answers at each of those links.