We do not necessarily recommend [2] (Hartshorne) as an introduction (or as a reference) to algebraic geometry. Even though [2] contains many exercises, some important results are among those exercises which are quoted in proofs of theorems. If one can solve those exercises by oneself, then [2] is a very useful book. The cohomology theory of coherent sheaves, including a proof of Serre's duality, is treated much more carefully in [2] than in other books. One can read [2] focusing on cohomology theory. In [2], only a projective morphism is considered in proofs, which does not cause any problems. It is more important to be able to use cohomology than to be able to prove cohomological results.
