I think Cox-Little-O'Shea's Ideals, Varieties, and Algorithms is a great text if you're looking at algebraic geometry for the first time. It focuses on the computational side of algebraic geometry and the commutative algebra behind it, so you won't hear about schemes, cohomology, etc., but it helps develop your intuition for algebraic geometry problems and skills in commutative algebra, and has loads of fun problems in every chapter. In particular, the chapter on projective algebraic geometry is awesome and has tons of interesting, instructive, and approachable exercises.