Apart from other excellent suggestions:
- 3264 and all that, David Eisenbud and Joe Harris.
Contains a wealth of useful information, such as a glimpse of intersection theory, geometry of Grassmanians, Chern classes, Hilbert schemes and a toolbox for Riemann-Roch.
- Lectures on Vector Bundles, J.Le Potier.
Concise introduction into Quot-shemes, moduli spaces of sheaves and stability conditions.
- The Geometry of Moduli Spaces of Sheaves, D. Huybrechts and M. Lehn
Another (more thorough) treatment of semistable coherent sheaves.