I'm looking for the conditions on $X$ so that the isometry group of $X$ is separable. We are taking the group to have the operation of composition and the topology of pointwise limits.
To be honest, I don't even know any isometries besides the identity function so I could possibly make due with some examples of isometries.
Any hints/answers are greatly appreciated.
No. Let $X$ be a countably infinite space with the discrete metric. Then any permutation of $X$ is an isometry. There are uncountably many, and there are no nontrivial convergent sequences of isometries. For example of isometries, these abound, depending on your space.
So for $\mathbb{R}^2$ you can have translations and rotations around the origin and reflection about a line and compositions of these.