There are (at least) two details you are missing.
(1) This is not an equivalence of categories between finitely generated graded modules and coherent sheaves. If your module is $0$ in all sufficiently large degrees, then the corresponding sheaf will be zero. For example, let $S=k[x,y]$ and $M=S/\langle x,y \rangle$. The sheaf on $\mathbb{P}^1$ corresponding to $M$ is the zero sheaf.
The category you want to work with is the one whose objects are finitely generated graded modules, and where we formally invert any map which is an isomorphism in sufficiently large degree. (Alternative formulation: we formally invert a map $f:M \to N$ if, for any $s$ in the irrelevant ideal, $s^{-1} f: s^{-1} M \to s^{-1} N$ is an isomorphism.)
(2) Not every coherent sheaf is a vector bundle. Correspondingly, not every finitely generated graded module will correspond to a vector bundle. If we were dealing with an affine variety, vector bundles would correspond to locally free modules. (Also called projective modules.) For projective varieties, things are a little trickier: the criterion is to be locally free away from the irrelevant ideal.
Sadly, I believe that there exist examples of modules which are locally free away from the irrelevant ideal, but are not isomorphic (in the above category) to any module which is locally free at the irrelevant ideal. This should be related to my question here. But you can go a long while without paying attention to this detail.
Best Answer
We can back up a step and ask about the relative spec of the symmetric algebra of any coherent sheaf. Then there are lots of nice examples. If $X=\mathrm{Spec}\ k[t]$ and $E$ is the skyscraper sheaf at $0$, then we get $\mathrm{Spec}\ k[t,u]/(tu)$. If $X$ is $\mathrm{Spec}\ k[x,y]$ and $E$ is the ideal sheaf of the origin, then I get the relative spec of $\mathrm{Sym}_X(E)$ is $\mathrm{Spec}\ k[x,y,t,u]/(xu-ty)$. This has a one dimensional fiber over the points of $X$ other then $(0,0)$, and a two dimensional fiber over $(0,0)$. It looks like the fiber over $x \in X$ is the dual vector space to $E_x \otimes_{\mathcal{O}_x} k(x)$, but I am not sure whether this is always true.
If you insist that $E$ be the dual sheaf to some sheaf $F$, then you run into the problem that non-locally-free dual sheaves are hard to come by in dimensions 1 and 2. See my question, to which there are several good answers.