Real algebraic geometry vs. algebraic geometry

Real algebraic geometry comes with its own set of methods. While keeping in mind the complex picture is sometimes useful (e.g. for any real algebraic variety X, the Smith-Thom inequality asserts that $b(X(\mathbb{R})) \leq b(X(\mathbb{C}))$, where $b(\cdot)$ denotes the sum of the topological Betti numbers with mod 2 coefficients), most of the technique used are either built from scratch or borrow from other areas, such as singularity theory or model theory.

The literature is a lot smaller for RAG than for traditional AG; the basic reference is the book by Bochnak, Coste and Roy (preferably the English-language edition which is more recent by more than 10 years, and has been greatly expanded). The book covers in particular the real spectrum, the transfer principle (which makes non-standard methods really easy), stratifications and Nash manifolds, among other topics. Michel Coste also has An Introduction to Semialgebraic Geometry available on his webpage a very short treatment of some basic results, enough to give you a first impression.

Other interesting books tend to be shorter and more focused than BCR, dealing with a specific aspect; e.g. Prestel's Positive polynomials. (dealing mostly with results such as Schmudgen's theorem), and Andradas-Brocker-Ruiz Constructible sets in real geometry (dealing mostly with the minimum number of inequalities required to define basic sets). The book by Benedetti and Risler is very interesting and concrete; I found some passages very useful and some results are hard to find in other books (the sections on additive complexity of polynomials are very thorough), but it is a bit scatterbrained for my taste.

As the name indicates, the book by Basu Pollack and Roy is entirely focused on the algorithmic aspects. It's a very good book, and you may still pick up some of the theory in there, but it does not sound like what you are after right now.

As for o-minimality, there again, Michel Coste's webpage contains an introduction that nicely complements van den Dries's book. I would hesitate to bundle o-minimality with real algebraic geometry. In some respects, the two domains are undoubtedly close cousins, and o-minimality can be seen as a wide-ranging generalization of real algebraic structures; on the other hand, each disciplines has also its own aspects and problems that do not translate all that well into the other.

I'm being verbose as usual. Still, I hope it helps.

There's also Partially Ordered Rings and Semi-Algebraic Geometry by Brumfiel and Real algebraic and semi-algebraic sets by Benedetti and Risler.

For an easy introduction to RAG, you could read van den Dries book "Tame topology and o-minimal structures": he treats the more general notion of o-minimal structures instead of real closed fields, and he does not uses any tool from AG.