Is every manifold a metric space? I'm trying to learn some topology as a hobby, and my understanding is that all manifolds are examples of topological spaces. Similarly, all metric spaces are also examples of topological spaces. I want to explore the relationship between metric spaces and manifolds, could it be that all manifolds are examples of metric spaces?

It depends on whom you ask. As a general topologist I would say no, there are manifolds (locally Euclidean (of a fixed dimension) Hausdorff spaces) that are not metrisable, like the long line. These can be quite interesting.
But in many cases, manifolds are assumed to be second countable (or some equivalent definition) and often also connected as well. And in such a context all manifolds are special metric spaces.
So it depends on the context. If you want differentiable manifolds (or smooth manifolds etc.) we almost always go into a setting where everything is supposed to be metrisable. And this is where they most often arise.
