In hyperbolic geometry, we know that any complete hyperbolic surface is the quotient $\mathbb{H}^{2}/\Gamma$ where $\Gamma < \text{Isom}(\mathbb{H}^{2})$ a discrete subgroup.
We know $\text{Isom}^{+}(\mathbb{H}) \cong PSL(2,\mathbb{R})$ which contains $PSL(2,\mathbb{Z})$ as a discrete subgroup. Also $\Gamma(m)$ which is the kernel of the natual map $PSL(2,\mathbb{Z}) \to SL(2,\mathbb{Z}/m\mathbb{Z})$ acts freely and properly discontinuously on $\mathbb{H}^{2}$ so $\mathbb{H}^{2}/\Gamma(m)$ is a hyperbolic surface. I want to see a geometric picture of this manifold. How can I proceed to see what this manifold look like (at least topologically, if not isometrically)?
This is related to very deep questions in number theory, and there's no simple answers.
From a naive point of view it seems reasonable. You can start with a fundamental domain for the fractional linear action of $PSL(2,\mathbb Z)$ on the upper half plane; the conventional fundamental domain is given by intersecting $-\frac{1}{2} \le \text{Real}(z) < \frac{1}{2}$ with $\text{abs}{z} \ge 1$. Then you can carefully choose coset representatives of $\Gamma(m)$ in $PSL(2,\mathbb Z)$, use them to translate the fundamental domain around, and hope that you chose those representatives carefully enough so that the union of those translated fundamental domains is a polygon which you can then glue up to get $\mathbb H^2 / \Gamma(m)$. For the case $m=2$ this can probably be done by hand. But the index of $\Gamma(m)$ in $PSL(2,\mathbb Z)$ grows uncomfortably fast: it is cubic in $m$. So this method is not practical in general.
The general heading of your question is the topic of congruence subgroups of $PSL(2,\mathbb Z)$. In that reference you'll see, for example, a link to a 2006 paper by Long, Machlachlan and Reid proving that there are only finitely many values of $m$ such that the quotient surface $\mathbb H^2 / \Gamma(m)$ has genus zero, meaning that it is homeomorphic to a sphere with some finite number of points removed. You'll also see described some amazing connections between the topology of $\mathbb H^2 / \Gamma(m)$ (more properly, not that surface itself but a closely related quotient orbifold) and the prime factors of the order of the monster group!