I'll give it a try (assuming you are referring to Griffths and Harris, Principles of Algebraic Geometry):
Chapter 1 without section 5,
Chapter 2-sections 1,6 and 7,
Chapter 3-without section 5
Chapter 4-without section 6
It think this covers a minimum background for string theory. Also you may want to look at C.Voisin's Hodge Theory and Complex Algebraic Geometry.