I am by no means an algebraic geometer, but have used many of its ideas and results for studying algebraic dynamics. I found that the book "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea was a wonderful way to learn both the algebra and geometry, all in the context of concrete computations using Groebner bases. This is a beautifully written text, quite accessible to undergraduates, filled with great examples and problems. Highly recommended.