One good source my undergraduate adviser recommended to me are the lecture notes of Candelas on Complex Geometry. They are written with string theory in mind and cover a lot of basic ground. I am not sure, if they are available online. Griffiths and Harris is very good, but probably not suitable as your only source for self-study. Just to get an idea what ideas were needed in string theory 25 years ago a look at chapter 12,14,15,16 in the second volume of Green, Schwarz, Witten might be helpful. Especially 14 and 15 should be interesting to you, even if you did not take a course in string theory yet.
By now there are of course a lot of other applications of ideas from algebraic geometry to the study of string theory beyond those ordinarily found in textbooks. For example model building in $F$-theory requires among other things to the study of singularities of elliptic fibrations and the approximate dynamics of certain branes is determined by variations of hodge structure. To actually find interesting examples knowledge of toric varieties is helpful. Most of those topics are actually not discussed in introductory texts.
This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user orbifold