Roughly speaking, the Hirzebruch-Riemann-Roch theorem gives a formula for the Euler characteristic of a vector bundle $E$ on a projective variety $X$ in terms of the Chern classes of $E$ and the Chern classes of the tangent bundle $T_X$. The Grothendieck-Riemann-Roch theorem is a stronger theorem which generalizes this to the relative case of a family of varieties $\mathcal X/S$; the Hirzebruch-Riemann-Roch theorem is the special case where $S$ is a point.
More generally, Chern classes are defined not just for vector bundles, but for any coherent sheaf: given a coherent sheaf, we can choose a resolution by vector bundles, and define the Chern classes of the sheaf by requiring that the Whitney sum formula hold for the exact sequence. The theorems are then naturally extended to this setting as well.
Now zero cycles can be associated with either their structure sheaf or their ideal sheaf, and we can perhaps compute the associated Chern classes by computing a resolution. Then the Hirzebruch-Riemann-Roch theorem will tell you the Euler characteristic of these sheaves.
Basically, the thing that makes the study of curves so easy is that points are codimension 1, so that the ideal sheaf of a point is actually a line bundle, and in particular locally free. When dealing with higher codimension objects, it is absolutely necessary to deal with non-locally free objects.