Something is quasi convex (concave) if the lower (upper) contour is convex. I don't know if we can talk about an upper/lower contour for a function, not a function's level curves/sets. Therefore, can we talk about a function be quasi-concave/convex, or must we be discussing level curves/sets?
For example, think of $y = \frac{1}{x}$ over domain $[0,\infty]$. To me it seems like this satisfies $f(\lambda x + (1-\lambda)y \leq \lambda f(x) + (1-\lambda)y$, where $f(\cdot) = \frac{1}{\cdot}$ (perhaps this is my problem; maybe I am getting 2 and 3+ dimensional properties confused?) which would mean it is convex, but it also satisfies $f(\lambda x + (1-\lambda)y \geq \min \{f(x),f(y)\}$, and is thus quasi-concave. Therefore this would be convex, which implies quasi-convex, and also quasi-concave, meaning it is quasi-linear...
I have to be making some mistake here; I am just wondering if someone can point it out to me
Yes, it makes perfect set to talk about quasiconvex and quasiconcave functions. Case in point: Wikipedia article Quasiconvex function. As you supposed, $f$ is quasiconvex if $\{ x:f(x)< a \} $ is convex for every $a$.
Caveat: the definition you have in mind is the one prevalent in optimization and assorted applications of mathematics. In calculus of variations, there is another property that goes by the same name; occasionally qualified as Morrey quasiconvexity.