Take a look at Harris Algebraic Geometry and Mumford Complex Projective Varieties. See how you fare with the material on the first pages. If it's alien to you, go learn the required stuff - complex analysis, differential forms, some manifolds and for Harris refresh a lot on that Linear Algebra and Projective Geometry. Hell, I would even start by reading some old classical projective geometry something like the book Plane Algebraic Curves from the Birkhauser editorial.
The thing is, you can go right away into Vakil or Hartshorne and learn schemes, but you will inevitably need to be know a lot of differential geometry, topology and complex analysis if you want to do serious Algebraic Geometry. So I would start by learning some more general math before diving into AG, and when the time comes for you to really go into AG, things will make MUCH MORE SENSE.
Oh, and don't forget, computations, lots of computations, don't let the overwhelming theoretical aspect of Algebraic Geometry deviate you from knowing how to explicitly calculate things like a blow up of a point