For example, if I drew $y=f(x)$, calculated its stationary points, roots and whatnot; is there a simple way I could draw the graphs of: $y=f(x^2)$, $y=f(\sqrt{x})$ and $y^2=f(x)$?
Also related to this, how do you go about drawing graphs of absolute value functions such as $y=f(|x|)$, $y=|f(x)|$ and some mixture of absolute values (e.g. $y=|x-3|+|x+7|$)?
Well, it depends primary on what is actually $f(x)$ answering to your first concern. For example, If $f(x)$ is a monomial like $x^2$ then $f(x^2)$ is the same as $\left(f(x)\right)^2$ but this is not the common case. You need to be more specific on which function are you looking to graph and the domain.
The absolutes values can be graphed in this way: say $\lvert x\rvert$ is your function then, if $x > 0$ the function is the same, but in $x<0$ then you have to add a minus sign to the original function inside of $\lvert \cdot \rvert$ like $-x$ in the case of $\lvert x\rvert$.
so $$ \lvert x\rvert = \begin{cases} \phantom{-}x & x \ge 0 \\ -x & x < 0 \end{cases} $$
When the function is a bit more complex, you need to separate it on cases, like I did with $\lvert x\rvert$. But you may need to take $x < 3$ and $x > 7$, and so on with all possible combinations.
Hope it helped