How do I show that if $X$ is simply connected, then there is a unique path class in $X$ with initial point $x$ and terminal point $y$?
Doesn't this just follow from the definition of simply connectedness?
It is not completely immediate. You want to show that if you have two paths $\gamma_1$ and $\gamma_2$ both from $x$ to $y$, then there's a homotopy from $\gamma_1$ to $\gamma_2$.
You can construct a closed loop as $\gamma_1-\gamma_2$ and (since the space is simply connected) let that contract to a point -- but in the intermediate stages of that contraction you have no guarantee that $x$ and $y$ will still be hit (indeed it cannot be the case that both are hit all the way, unless $x=y$). So you need to do some plumbing and splicing in order to derive a homotopy between $\gamma_1$ and $\gamma_2$ from the null-homotopy. For complete rigor, showing the homotopy as an explicit function defined by a case split is probably in order.