It has been estimated that, at the present stage of our knowledge, one could give a 200 semester course on commutative algebra and algebraic geometry without ever repeating himself. So any introduction to this subject must be highly selective.

--E. Kunz, Introduction to commutative algebra and algebraic geometry, 1978.

Overview

This semester, Math 673 will be an introduction to schemes, with an emphasis on arithmetic applications. Schemes provide a unified framework for extensions of classical algebraic geometry, such as:

Working over fields like Q or F_p, instead of insisting on an algebraically closed field;
Tangent spaces, deformations, and other infinitesimal calculations;
Reducing an abstract variety modulo p;
Relating generic behavior to special behavior;
Exploiting the parallels between number fields and function fields of curves;
etc., etc., etc.

One of the main motivations for constructing the theory of schemes was to prove Weil's conjectures on the number of solutions to systems of equations over finite fields. The course will conclude with an overview of Deligne's proof of these conjectures and some of their applications.

Textbook

Algebraic geometry and and arithmetic curves, by Qing Liu is a fantastic reference, although perhaps a little hard to read.
The geometry of schemes, by David Eisenbud and Joe Harris, is a wonderful read, but probably wouldn't make a very good reference book.