Looking for an introductory Algebraic Geometry book I am looking for recommendations on an AG text to work through this summer, possibly with the help of a mentor. I would want this book to have some introduction to categories, and then develop the modern methods (some development of sheaves and schemes), hopefully up to some realization of Riemann-Roch. Any recommendations? 

These notes by Andreas Gathmann are precisely what you're asking for. He starts very gently, schemes being introduced in Chapter 5, but he ends with sheaf cohomology (including Riemann-Roch) and some intersection theory.
