Cox Little and O'Shea's "Ideals varieties and algorithms" (http://www.amazon.com/Ideals-Varieties-Algorithms-Computational-Undergraduate/dp/0387356509/ref=sr_1_1?ie=UTF8&s=books&qid=1265456210&sr=1-1) is very accessible, assumes almost no background in commutative algebra, and has many examples. The emphasis is on computational algebraic geometry (including Groebner bases).
BTW, Milne's "Algebraic Geometry" (http://jmilne.org/math/CourseNotes/AG.pdf) includes an "Annotated Bibliography" Appendix with an "Elementary Algebraic Geometry" section, and perhaps this is a good place to start the search.
