Equation of a Riemann surface? Intuitively in complex analysis I know what a Riemann surface is. It is a surface such that at every point on it the value of a function $f(z)$ is single-valued. However, how would I go about finding the equation for such a surface, i.e. if I was to plot one what one I plot?

When you are talking about the equation of a surface you have equations like $x^2+y^2+z^2=1$ for a sphere or $z=x^2+y^2$ for a paraboloid in ${\mathbb R}^3$ in mind. But this is not the appropriate way to think about a Riemann surface. A Riemann surface can be any orientable two-dimensional manifold, i.e.,  "surface" in the intuitive way. In fact the surfaces I mentioned can be the bottom carrier of various Riemann surface structures, whereby the latter will  then differ in their  local or global conformal structure. To put it bluntly: A Riemann surface is an orientable $2$-manifold on which an angle measurement is defined on all tangent planes.
The surfaces I mentioned inherit an  angle measurement in their tangent planes from their imbedding in ${\mathbb R}^3$, whereas the "Riemann surface of the function $z^2=w$" inherits its angle measurement from the angle measurement in the complex $z$-, or $w$-plane (the point $z=w=0$ is special).
To sum it up: A Riemann surface is an orientable $2$-manifold on which you have introduced local coordinates $z_\alpha$ $(\alpha\in I)$ such that in intersections of coordinate patches you have $z_\alpha=\phi_{\alpha\beta}(z_\beta)$ with analytic transition functions $\phi_{\alpha\beta}$.
