I don't think Hendrik's recommendation is particularly helpful for an absolute beginner in algebra.
Here's how I'd do it:
Let $X$ denote a vectorspace and $S$ denote a subspace. We want to show that the canonical projection $X \rightarrow X/S$ is surjective.
Recall that $\mathcal{P}(X)$ is notation the collection of all subsets of $X$; this is called the powerset of $X$.
Now lets just try to remember the definitions. Let $\pi : X \rightarrow \mathcal{P}(X)$ denote the function given as follows: $$\pi(x) = x+S$$
The quotient $X/S$ is, by definition, the image of $\pi$, and the canonical projection $X \rightarrow X/S$ is, by definition, the co-restriction of $\pi$ to its image. But since every function is surjective when co-restricted to its image, hence the canonical projection is surjective.