The book by Kreuzer and Robbiano provides an excellent hands-on introduction to the field of Computational Commutative Algebra.
I am currently teaching a course "Topics in Applied Mathematics" at UC Berkeley with emphasis on Solving Polynomial Equations, and the book "Computational Commutative Algebra 1" was at the top of reading list I gave to my students. It provides them with a solid foundation for more advanced study and research in algorithms for algebraic geometry.
Bernd Sturmfels