$5$-vertex graphs with vertices of degree $2$ I'm trying to show that all graphs with $5$ vertices, each of degree $2,$ are isomorphic to each other.  Is there a more clever way than simply listing them all out?

I'm assuming you do not allow multi-edges, as otherwise there is a trivial counterexample (the cycle of length $5$, versus a disconnected graph consisting of a triangle and two vertices joined by two edges).
Pick one vertex $v_0$; it must be joined to two other distinct vertices, which we may call $v_{-1}$ and $v_1$. Each of those must be joined to another vertex; can they be joined to each other? Can they both be joined to the same vertex that is not yet listed? Consider the possibilities. Then see where each of them leads you. 
