Every linear mapping satisfies $f(0) = 0$ In my textbook is written that every linear mapping $ f: V \to V'$ satisfies $f(0)=0$. But what about the mappings between the polynomials like $p(x) = ax^2 + bx + c$, where $c$ is nonzero? Elementary probably, but fundamental.

There is a null polynomial $p(x) = 0$. What you mean is an affine space, not a linear space.
About the property:
Every vector space has a null vector $\vec{0}$ so for a linear map $f$ we have
$$
f(\vec{0}) = f(0 \, \vec{0}) = 0 f(\vec{0}) = \vec{0}
$$
