In addition to the excellent text by K. Smith et al. recommended above, you could consider the book
Ideals, Varieties, and Algorithms by David A. Cox, John B. Little and Don O'Shea
which is very accessible, and some older texts which are more elementary and (therefore) quite readable, e.g.
-- Rudiments of algebraic geometry by William Elliott Jenner
-- Introduction to algebraic curves by Phillip A. Griffiths