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-schemes, 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. 

 - *Fundamental Algebraic Geometry. Grothendieck's FGA explained*, B. Fantechi et al. 

An introduction into some more advanced fundamental techniques: descent theory, Hilbert and Quot schemes, elementary deformation theory and Picard scheme.