Is there a general method for calculating things like $\operatorname{Spec} \mathbb{C}[x,y]/I$ ?
Maximal ideals are $ \{(x-\tilde{a},y-\tilde{b}): b^2-a^3=0\}$ because of $f(x,y)(x-a)+g(x,y)(x-b)=y^2-x^3$ evaluated in $x=a$ and $y=b$. But how to calculate non-maximal ideals?
Just a hint. $\operatorname{Spec}\mathbb{C}[x,y]/I$ is in bijection with the set of all primes $p$ in $\mathbb{C}[x,y]$ such that $I(=(y^{2}-x^{3}))\subseteq p$. Now, can you describe what are the elements of $\operatorname{Spec}\mathbb{C}[x,y]$? (Hint: you have the zero ideal and the maximal ideals in that spectrum and certainly all the principal ideals generated by an irreducible polynomial in $\mathbb{C}[x,y]$. Prove that these exhaust all the possibilities).