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?

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$As worded the question is a bit awkward to address.
Abstractly, a Riemann surface is a smooth real surface equipped with a holomorphic structure, i.e., a holomorphic curve. Examples include open subsets of $\Cpx$, the complex projective line $\Cpx\Proj^{1}$, tori $\Cpx/\Lambda$ with $\Lambda \subset \Cpx$ a rank-two lattice, and hyperbolic quotients (which tend to be difficult to describe explicitly).
Concretely, certain multi-valued complex functions defined on (open subsets of) the plane, such as logs and roots of entire functions, are associated with Riemann surfaces, and may be understood as subsets of $\Cpx\Proj^{1} \times \Cpx\Proj^{1}$, or as multiple copies of the complex line $\Cpx$ with suitable branch cuts and cross-gluings and added points at infinity. Examples include "the Riemann surface of the $n$th root", $w = \sqrt[n]{z}$ or $w^{n} = z$; "the Riemann surface of $\log$", $w = \log z$ or $\exp w = z$; and "the circle" $w = \sqrt{1 - z^{2}}$, or $z^{2} + w^{2} = 1$.
Alternatively, a homogeneous polynomial $f$ in three variables (whose gradient vanishes only at the origin of $\Cpx^{3}$) defines a Riemann surface embedded in the complex projective plane $\Cpx\Proj^{2}$.
All three classes are difficult to represent geometrically with complete fidelity: Abstract Riemann surfaces are not embedded in anything (and don't have an intrinsic "equation" in the sense you mean), while the concrete "plane curves" naturally sit inside a (real) four-dimensional ambient space, and inevitably undergo distortion (acquire singularities and/or self-intersections) when projected into real three-dimensional space.
That said, a particularly nice example is Costa's surface, a minimal surface in $\Reals^{3}$ of a torus with three punctures given by its Weierstrass embedding.
