I apologize in advance for the vagueness of my question, but I am looking for sources (if they exist) where $K_0$ (the Grothendieck group of coherent sheaves) is computed "by hand" for some low-dimensional varieties (besides nonsingular curves, which is an exercise in Hartshorne).

To try to explain further, here is an example of what I am not looking for: in the case of toric surfaces, one can obtain the $T$-equivariant $K$-theory by localization, and then mod out the equivariance to obtain regular $K$-theory. This does not give me enough understanding about what relations in the Grothendieck group are really at play.