MR1075991 Abhyankar, Shreeram S. Algebraic geometry for scientists and engineers. AMS Providence, RI, 1990. It is really for scientists and engineers.

Another excellent choice is D. Cox J. Little and D. O'Shea,
Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics. Springer, New York, 2007.

Griffiths Harris is far too difficult. Shafarevich is OK for a beginner but beginner-mathematician.