Two things. I'll talk about SL_n rather than GL_n, but that's just a technicality.

- There is a short exact sequence

$1 \rightarrow SL_n(\mathbb{Z},m) \rightarrow SL_n(\mathbb{Z}) \rightarrow SL_n(\mathbb{Z}/m\mathbb{Z}) \rightarrow 1,$

where $SL_n(\mathbb{Z},m)$ is the level $m$ congruence subgroup of $SL_n(\mathbb{Z})$. There is a nice presentation $\langle S|R \rangle$ for $SL_n(\mathbb{Z})$ due essentially to Magnus; it can be found in Milnor's book on algebraic K-theory (look for the chapter that calculates $K_2(\mathbb{Z})$). The generating set $S$ is just the set of elementary matrices. To get from there to $SL_n(\mathbb{Z}/m\mathbb{Z})$, you just need to add a normal generator for $SL_n(\mathbb{Z},m)$ to $R$. By the solution to the congruence subgroup problem for $SL_n(\mathbb{Z})$ (due to Mennicke, but you are better off looking at Bass-Milnor-Serre's paper), this congruence subgroup is normally generated by the mth power of a single elementary matrix.

A similar idea also works for the symplectic group. To find a presentation for $Sp_{2g}(\mathbb{Z})$, look at Theorem 9.2.13 of Hahn and O'Meara's book "The Classical Groups and K-Theory".

- I also want to make a comment on DLS's answer. There is a related short exact sequence

$1 \rightarrow V \rightarrow SL_n(\mathbb{Z}/p^{k+1}\mathbb{Z}) \rightarrow SL_n(\mathbb{Z}/p^k\mathbb{Z}) \rightarrow 1.$

The group $V$ has a beautiful description, due essentially to Lee and Szczarba. Namely, it is isomorphic to the additive group underlying the special lie algebra over $\mathbb{Z} / p\mathbb{Z}$ (in particular, it is abelian). Moreover, the action of $SL_n(\mathbb{Z}/p^k\mathbb{Z})$ on $V$ factors through the adjoint representation of $SL_n(\mathbb{Z}/p\mathbb{Z})$ on the special lie algebra.

Lee and Szczarba did not write this down in quite this form, but I wrote out a similar result for the symplectic group in Lemma 3.1 of my paper "The Picard group of the moduli space of curves with level structures".