Currently reading a couple of texts on this. One that has become a classic, I think, is Herman Weyl’s ‘Concept of a Riemann Surface’.
This was originally a series of lectures from the early 1900s that evolved with new developments in Topology in the 30s-50s.
There are a lot of resources nowadays though. Online lectures and study guides are just as helpful, imho.
A Riemannian Manifold is defined as something different - roughly, a smooth manifold endowed with a metric.
Re: Riemannian Manifolds, there’s an awesome three-part series by Lee (I think??). Starting with Topological Manifolds then Smooth Manifolds then Riemannian Manifolds. It’s kind of funny, in pure math, Riemannian manifolds seem to be toward the end of the series, in physics I think you have to get to Riemannian manifolds pretty quickly otoh. Not a physics guy tho, so don’t quote me on that.