For instance, say $G = \langle x , y \ | \ x^{12}y=yx^{18} \rangle$. I want to know what is the normal closure of $y$ in $G$.
In general, what are the standard approaches to compute the normal closure of a subset of a finitely presented group? Are there algorithms?
You can compute the normal closure by computing the quotient, and then considering the kernel of the quotient homomorphism.
For the example you gave, let $N$ be the normal closure of $y$ in $G$. Then $G/N$ has presentation $$ \langle x,y \mid x^{12}y = yx^{18},y=1\rangle $$ This presentation reduces to $\langle x \mid x^{12} = x^{18}\rangle$, which is the same as $\langle x \mid x^6 = 1\rangle$.
Thus $G/N$ is a cyclic group of order 6, and $N$ is the kernel of the homomorphism $G\to G/N$. In particular, the normal closure of $y$ consists of all words for which the total power of $x$ is a multiple of 6.
