The book Geometric Algebra For Computer Science, by Dorst, Fontijne, and Mann has one of the best introductions to the subject that I have seen.

It contains particularly good introductions to the dot and wedge products and how they can be applied and what they can be used to model. After one gets comfortable with these ideas they introduce the subject axiomatically. Much of the pre-axiomatic introductory material is based on the use of the scalar product, defined as a determinant. You'll have to be patient to see where and why that comes from, but this choice allows the authors to defer some of the mathematical learning overhead until one is ready for the ideas a bit better.

Having started study of the subject with papers of Hestenes, Cambridge, and Baylis papers, I found the alternate notation for the generalized dot product (L and backwards L for contraction) distracting at first but adjusting to it does not end up being that hard.

This book has three sections, the first covering the basics, the second covering the conformal applications for graphics, and the last covering implementation. As one reads geometric algebra books it is natural to wonder about this, and the pros, cons and efficiencies of various implementation techniques are discussed.

There are other web resources available associated with this book that are quite good. The best of these is GAViewer, a graphical geometric calculator that was the product of some of the research that generated this book. Performing the GAViewer tutorial exercises is a great way to build some intuition to go along with the math, putting the geometric back in the algebra.

There are specific GAViewer exercises that you can do independent of the book, and there is also an excellent interactive tutorial available. Browse the book website, or Search for '2003 Game Developer Lecture, Interactive GA tutorial. UvA GA Website: Tutorials'. Even if one decided not to learn GA, using this to play with the graphical cross product manipulation, with the ability to rotate viewpoints, is quite neat and worthwhile.