To what branch of mathematics does the Hodge conjecture belong? I'm aware that it's very advanced, but what kind of prerequisites would one need to understand those problems? Can you suggest some good texts for a senior undergraduate/beginning graduate?
The Hodge Conjecture belongs to Topology and Geometry. Courses like Introductory Topology, Differential Geometry, Algebraic Topology, Riemannian Geometry, Complex Manifolds, and Differential Topology will be critical to understanding the mathematics necessary for Hodge Theory and the Hodge Conjecture. If it is your goal to prove the Hodge Conjecture (or disprove it), as is my goal, you will also want to make an advanced study of formal proofs. In addition to the Voisin text written above, I suggest:
Modern Geometries by James Smart. Algebra by Thomas W. Hungerford. Algebraic Topology by Allen Hatcher. Topology and Geometry by Glen E. Bredon. Riemannian Geometry and Geometric Analysis by Jurgen Jost. Differential Forms and Applications by Manfredo P. Do Carmo. And Introduction to Smooth Manifolds by John Lee.
Some additional advice not included in your original question: After completing a B.A. or (preferably) a B.S. in Mathematics, you will have to get accepted into a good mathematics graduate program. Your current academic advisor can explain that in more detail, but you will need to take the Graduate Record Examination (General), and the GRE (Mathematics), apply to the school(s) of your choice, and secure funding (i.e. student loans, fellowships, or through independent sources). Once that's done, you'll want to talk to your graduate academics advisor and/or doctoral advisor about the specific appropriate path of courses to take.