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

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