The title of the question already says it all but I would like to add that I would really like the book to be more about geometric algebra than its applications : it should contain theorems' proofs. Just adding that I have never taken a course on geometric algebra. I'm a 2nd year engineering student, so a "beginner" book style will be very good!!! Also mentioning what would be the prerequisites for mastering the branch is appreciated. Thanks.
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.
If you are really a beginner, I can recommend this book Geometric Algebra: An Algebraic System for Computer Games and Animation It has nothing to do with computer animations or games but describes the whole machinery of GA in a clear and CONCRETE manner without obscure formulas and with tables for all products in G2 and G3. He also explains how to calculate intersections of every objects from the standpoint of the usual methods and then shows how to do this in GA, I have never seen this so easily explained in any other book or literally in every pdf I found on the internet. It is also a short book with about 200 pages that can be read in one day. It doesn't cover conformal GA and GA calculus which is a good thing because as a beginner you don't understand it without being familiar with the basics. There are no exercises, but from the examples he gives you can easily make up your own. The book also doesn’t use a theorem-proof style concept (actually I think there is only one proof), but again I think as a beginners this is not so important. This should be the standard book for people who want to learn it ! Additionally to that I would recommend that you use the CluCalc Geometric Algebra software for calculating and visualising, it's very easy to learn, for an introduction you can use Google to search for pdfs about it.
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.
I believe that paper meets your criteria for containing the theorems and proofs, as well as a good collection of references.
This is a late answer, but the book in question wasn't published until 2019.
In my opinion, by far the best book about Clifford algebras and related things is Geometric Multivector Analysis: From Grassmann to Dirac by Andreas Rosén (and I'm not saying that just because I happen to know the author). It is aimed at advanced undergraduates, so it's maybe not completely “beginner style”, but he puts a lot of effort into explaining the ideas and the philosophy of the subject, and there are complete and crystal clear proofs of (almost) all the theorems. The book builds the theory from the bottom, starting with exterior algebras and Clifford algebras, and finishing with the Atiyah–Singer index theorem.
There is the classic book "Algèbre géométrique" ("Geometric algebra"), from E. Artin -if this is the subject about what you're asking. I only know the French version, but it should exist an English translation. About the prerequisites, I think that a first course on Linear Algebra would suffice.
A recent book, exceptionally clear and rigorous, is: Jayme Vaz Jr & Roldão da Rocha Jr (2016) An Introduction to Clifford Algebras and Spinors. It gives a thorough grounding for geometric algebras in a way that builds the concepts from the ground up, but does not make the mistake of "hiding" the inner product in the definition and does not define misbehaved operations such as Hestene's "dot product of multivectors".
This book is readable and hence suitable for someone at a 2nd year level, but it is helpful if you are comfortable with the concepts such as vector spaces without magnitudes (a way of comparing lengths of vectors that are not parallel) and the dual of a vector space.
I found “Linear and Geometric Algebra” by MacDonald exceptionally clear. https://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonald/dp/1453854932/ref=mp_s_a_1_1?crid=NM8K3HHV9PCO&keywords=linear+and+Geometric+Algebra&qid=1694625026&sprefix=linear+and+geometric+algebra%2Caps%2C424&sr=8-1