It seems that in the decades following the publication of Hartshorne's Algebraic Geometry, a sufficient number of aspiring mathematicians have run aground in efforts to digest the field, such that for many, an automatic response to a cry for help is to redirect the student to Mumford's "Red Book". I have located a few online alternatives, for those who cannot stomach the cost of yet another mathematics book.
Matt Kerr and Andreas Gathmann (independently) provide lecture notes on Algebraic Geometry that contain a more-than-average amount of motivation through pictures:
Algebraic Geometry III/IV - Matt Kerr
Algebraic Geometry - Andreas Gathmann
Why should one study Algebraic Geometry? Well, to solve problems in Algebraic Geometry, of course! However, in all seriousness, it seems that the advantages of studying Algebraic Geometry far out-weigh the disadvantages.
Unlike most other fields, it seems that the problems that modern Algebraic Geometry has solved, and the correspondences it has uncovered, are yet to trickle down to mathematical expositions at the more elementary level, resulting in a complete blank for the uninducted, at the outset.
