So, I have that the mgf of a r.v. X exists for all real values of t and is given by
$$M(t) = \frac{e^t-1}{t} \quad t\neq 0 \quad M(0) = 1$$
and I have to find the mean and variance of X.
My main problem is that $t$ in the denominator which I don't know how to expand. I'm using series expansion for $e^t$ and I get $$1 + t + \frac{t^2}{2!} + ... - 1 = t + \frac{t^2}{2!} + ...$$ but don't know what to do next.
You don't have expand the $t$ in the denominator. You know that: $e^t-1= t + \frac{t^2}{2!}+\cdots$. So if you divide that by $t$, you will have your expansion.