In addition to the already mentioned
Procesi, Lie groups. An approach through invariants and representations
which provides an excellent update on both Weyl's "Classical groups. Their invariants and representations" and the relevant parts of Hodge and Pedoe's "Methods of algebraic geometry", I'd like to mention an amazing book by Kraft, unortunately not translated into English yet (there is a Russian translation):
Kraft, Geometrische Methoden in der Invariantentheorie
For a bird's eye view, I recommend the following survey in Russian (Yellow Springer) Math Encyclopaedia:
Vinberg and Popov, Invariant theory, Algebraic geometry IV