Could you recommend a rich, clear, and complete theory book on euclidean, affine and projective spaces (i.e., "geometry"); and an interesting exercise book full of non-trivial problems and exercises?
Here are some Textbooks with this kind of geometry:
Dieudonne - Linear algebra and geometry
Vinberg - A course in algebra, chapter "affine and projective spaces"
What a coincidence! I am studying the same. I am following different books, each with a distinctive flavor. Although I am not sure what would constitute a complete book, here are my picks:
1) I started out with Michele Audin's Book entitled "Geometry". Her emphasis is on linear algebra. It is a 'theorem-proof-theorem-proof' type of a book. So, I felt that book was dry. This is not a criticism of her book, but it is about my reading style. I like historical motivations. It has beautiful exercises. She makes you derive most of the classical theorems.
2) Felix Klein's 'Elementary mathematics from an advanced standpoint-Geometry' is a REAL treat to read. It starts from completely elementary considerations and builds a lot of theory with nice stories and physical intuition. He further lays foundations for 'theory of invariants' towards the end. Be warned, you will not find exercises in this book.
3) Another great book, I believe, is 'Geometry' by Brannan,Esplen,Gray. It gives you both synthetic and analytic flavors of geometry. They explain things out in clear detail. The illustrations and presentations of the topics are awesome. This is the book I go to if I want everything spelt out in detail.
Finally, I am sure all the books by Coxeter do not need any introduction.