Good introductory book on geometric algebra

The classic reference is David Hestenes' New Foundations for Classical Mechanics which is by one of the early developers of geometric algebra.

You may find it easier to learn geometric algebra from Geometric Algebra for Physicists by Doran and Lasenby though (I certainly did). The link is to a sample version of chapter 1.

A reference that I've never looked at is Geometric Algebra for Computer Science which details the geometric algebra approach to computer graphics, robotics and computer vision.

As for prerequisites - certainly some familiarity with linear algebra. For the 'geometric calculus' component a first course in multivariable calculus would be sufficient. Since the big developments in geometric algebra in the 1980s were by physicists, many of the examples tend to be physically motivated (spacetime algebras, relativistic electrodynamics etc) and a passing familiarity with (special) relativity, rigid body dynamics and electromagnetism would be useful (though certainly not essential).

The previously mentioned "Geometric Algebra for Computer Science" is a good introduction that concentrates on the algebraic (not calculus related part) of GA. It does have material on GA's application to computer graphics, but the bulk of the text is just on the geometry behind GA.

Another possible starting point is "Linear and Geometric Algebra" by Alan Macdonald. This is a text that replaces the standard material of a first Linear Algebra course with the same topics using GA. It is, in my opinion, a great way to learn both Linear Algebra and Geometric Algebra. The text is developed rigorously with theorems and proofs but includes ample examples and motivation. This book also does not try to develop the calculus part of GA.

There are different meanings of the words Geometric Algebra.

One is represented by Artin's book on the reconstruction of algebraic structures (fields, rings) from the geometries that they coordinatize.

The other is the use of Clifford algebras, quaternions and related ideas as a formalism for geometry and physics. This is popularized by Hestenes and is somewhat controversial, not because the math is wrong, but because it uses extra metric structure in cases where not logically required, and because of the tendency to rename known concepts and overstate the differences and advantages compared to the conventional approach. Using quaternions to represent three dimensional rotations is not controversial at all and is an important method in computer graphics, but this is a different theme of much more limited scope than Hestenes' program to rewrite physics in Clifford algebraic language.

Besides the books by Hestenes, Hestenes and Sobczyk, Dorst, Doran and Lasenby, Porteous, Lounesto, and Baylis, you should find a rather accessible paper by Eric Chisolm on ArXiv.org. You will find its abstract at the following URL.

http://arxiv.org/abs/1205.5935

I believe that paper meets your criteria for containing the theorems and proofs, as well as a good collection of references.