What are some tools -- either theoretical/by hand or algorithmic/by computer -- that are useful for doing computations in finitely presented groups?

In my particular case, I'm working with a finitely presented group $G=Mod(S_{0,n})$, the mapping class group of an $n$-punctured sphere. Inside there I have a subgroup $H \approx \mathbb{Z}^2$. After adding some relations to $G$ -- namely, killing high powers of the half twist Artin generators -- I expect that the image of $H$ in the quotient should now have rank less than 2. It would be great to actually have in my hand an element of $H$ which is killed by these new relations, and I doubt that guessing words in the generators of $H$ and fiddling with relations for hours by hand would be very productive.

If folks have general suggestions about tools they use for these sorts of things (not just for my question in particular), that would also be useful.

somesense of what kind of group the quotient of $G$ is. $\endgroup$