I was wondering if there is a sure fire process of finding the ranges of functions without plotting them on a graph?
At the moment (only for quadratics) I find the vertex of the graph through $(\frac{-b}{2a},f(\frac{-b}{2a}))$ then if $a$ (the first term in the function) is greater than zero, the parabola is open upwards and vice versa. This means that the y cannot be less than the vertex number, and can go to infinity if domain goes to -infinity or infinity?
Is this a logical way to think about quadratics? And is there any other mathematical way of finding the ranges of other functions that aren't quadratics?
Yes, and No.
For quadratics, it is easy : They are all parabolas, so the only relevant things you need to know are the vertex and whether it is concave up or down. Knowing that, you can find the range by imagining the graph.
For a general function, there is no catch-all procedure. However, one that works is to "solve for x". ie. Given $f$, try to find a function $g$ such that $f(g(x)) = x$ for all $x$. If such a function $g$ exists, then the range if $f$ is everything.
Identifying the obstructions to finding such a $g$ will eventually help you find the range.