I am editing this question so that it is clearer for other users with a similar question, although I consider the answers I received sufficiently explanatory for my purposes.
I am an undergraduate, and I have a hard time drawing a connection between group presentations and group actions. So, is there an explicit way to take a group presentation and derive the action of the group from this presentation?
For example, let's say you were given the presentation $\langle x,y| x^n = y^2 = e, yx=x^{-1}y\rangle$, which is the dihedral group $D_{2n}$. Is there an algorithmic or heuristic way to derive the possible actions of $D_{2n}$, or any group $G$, from its presentation, such as determining a homomorphism $G\longrightarrow S_n$?
There are algorithms to solve the problems that you are describing which have efficient implementations in GAP and Magma. In particular, to find the transitive actions of a group defined by a finite presentation of degree up to a specified number $n$, you can use the $\mathtt{LowIndexSubgroups}$ algorithm, which finds representatives of conjugacy classes of subgroups of index up to $n$.
But I am guessing that this is not what you are looking for. A very naive method of finding the homomorphisms of your finitely presented group $G$ to $S_n$ is to try all elements of $S_n$ as the images of each generator of $G$. For each such assignment of images of generators, you can check whether this extends to a homomorphism of $G$ by checking whether the images satisfy the relations of $G$. But this is impractical except for very small $n$. To find homomorphisms of $D_{2n}$ to $S_n$ you would need to try $(n!)^2$ possible generator images. This can be significantly reduced by using a bit more theory - as a first improvement, for the images of the first generator you need only try representatives of conjugacy classes of $S_n$, and with further refinements of that type the method would become feasible for a computer but still impractical to do by hand.
I suspect that this will not be very helpful, but I am just trying to answer the question you asked.