I'm a number theorist, not an algebraic geometer, but I was trying to figure out an equation for $X_0(3)$ from scratch using my low-level techniques. I first wrote out the $q$-expansions for $j(\tau)$ and $j(3\tau)$ to get the modular polynomial $\begin{align*}P(x,y)=1855425871872000000000 x &+ 452984832000000 x^2 + 36864000 x^3 + x^4\\ + 1855425871872000000000 y&- 770845966336000000 x y + 8900222976000 x^2 y\\ - 1069956 x^3 y &+ 452984832000000 y^2 + 8900222976000 x y^2 \\ + 2587918086 x^2 y^2 &+ 2232 x^3 y^2 + 36864000 y^3 - 1069956 x y^3\\ &+ 2232 x^2 y^3 - x^3 y^3 + y^4.\end{align*}$
So we get the function field of $X_0(3)$ is the quotient field of $\mathbb{C}[x,y]/(P(x, y))$. The trouble with the modular equation, though, is that it's singular in a few places; to me, this means that the ring above isn't integrally closed, and the element $\begin{align*}t=&(y(-54000 + y) (-8000 + y)(32768 + y) (12167000000 - 52250000 y + y^2)\\ &(94878058000 - 52250000 y + y^2) (-681472000 - 1264000 y + y^2)\\ &(6774250000 - 1264000 y + y^2)(-134217728000 + 117964800 y + y^2)\\&(7728080400 + 117964800 y + y^2))/((x - y) (-52250000 + x + y)\\&(-1264000 + x + y) (117964800 + x + y))\end{align*}$
is a (quadratic) integer over the ring (all the factors on bottom correspond to singular $(x, y)$, and the factors on top cancel out the singular and nonsingular poles on the variety), so we get a quadratic extension $\mathbb{C}[x,y,t]/(P(x, y), (\cdots)t-(\cdots), t^2-(\cdots)t-(\cdots))$.
I think this ring is integrally closed, so how do I go from this (assuming I know the explicit quadratic expression, which is disgusting) to some nice genus $0$ formula? I think there's an indirect way to calculate it, using the $\eta$ function and $(\eta(3\tau)/\eta(\tau))^{12}$, but I'm a bit stubborn, and since continuing in this path has to work, I want to see it through.