When I first studied real algebraic geometry. I was more concerned with the algebra (probably even now). Similar to the classical algebraic geometry where students get to first learn commutative algebra before they do some algbraic geometry, I learned first real algebra (without the geometry). Real algebra alone is a big field and by the time I started real algebraic geometry it was a little late (so I practically did only real algebra during my PhD years). Still, if you do want to get the fundamentals of real algebra (before doing real algebraic and analytic geometry) and if you know some German, I would highly recommend the book of Knebusch and Scheiderer also available for free here. You will learn more about convex valuations, preorderings, partial orderings, real closed fields, cones, Artin Schreier Theorem etc.theorem, the real spectra, the Harrison topology and constructible topology, than you would in a typical commutative algebra book. If you don't know any German then of course you could still read BCR (Bochnak, Coste and Roy). My PhD thesis has a preliminaries section that covers some basics you would need to go through. I could also recommend Brumfiel Partially ordered rings and semialgebraic geometry (which I fond sometimes more helpful than BCR).. and of course one could not forget the introductory material written by T.Y.Lam An introduction to Real Algebra (Rocky Mountain Journal of Mathematics, 1984, Vol.14, No.4, p.767-814).
