I suggest "Introduction to Algebraic Geometry" by Brendan Hassett. It is a nice, down-to-earth introduction to algebraic geometry, and it also spends a lot of time on computational topics like Grobner bases. When I was an undergraduate, I took a class from Brendan using an early version of this book (in fact, I think he was writing it as he taught), and it was completely accessible despite the fact that at time I did not know much commutative algebra.
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