I'm only an "algebraic geometry enthusiast", so my advice should probably be taken with a grain of salt. With that said, here are some nice things to read once you've mastered Hartshorne.
1) I'm a big fan of Mumford's "Curves on an algebraic surface" as a "second" book in algebraic geometry.
2) Fulton's "Toric Varieties" is also very nice and readable, and will give access to some nice examples (lots of beginners don't seem to know enough explicit examples to work with).
3) More stuff about algebraic curves. The best book here would be "Geometry of Algebraic Curves" by Arbarello, Cornalba, Griffiths, and Harris. The next step would be to learn something about the moduli space of curves. An inspiring choice here would be "Moduli of Curves" by Harris and Morrison.
4) Intersection Theory. Fulton's book is very nice and readable.
5) Algebraic groups. I'm a big fan of Springer's book here, though it is written in the language of varieties instead of schemes.
EDIT : I forgot to mention Kollar's book on resolutions of singularities. A masterpiece of exposition!