There is a new book by Lakshmibai and Raghavan called Standard Monomial Theory which is mostly about how to do invariant theory in a "Schubert varietiesque" way. It is introductory to both Schubert varieties and invariant theory. I haven't read all of it, but I would recommend it because it works out a lot of different cases. I don't think you need that much algebraic geometry to read it. They review the relevant facts beforehand.
edit: I should also mention that one advantage of this book over Fulton (which I also recommend) is that it covers Schubert varieties in other types of Grassmannians. Also, it gives a nice application of Schubert varieties to invariant theory (in particular, how to calculate rings of invariants in a characterstic-free way in various cases of classical interest).