How to calculate the first Chern class (concrete simple example)?

Question

I have been finding a good source on computing first Chern class of a manifold (e.g.Riemannian) but I just couldn't find a single example of how to compute in terms of local coordinates. I wanted to understand what does it represents, and when it is positive, zero or negative what does these means ? Deeply appreciated if someone could provide a detail calculation to get the first Chern class of a simple manifold. I just know that it is a cohomology class.