Book suggestions on projective geometry
For a modern approach, see Richter-Gebert's
Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry.
A nice interesting book which has a couple of chapters at the start on Projective Geometry, and really the applications of it in Algebraic Geometry is Miles Reid's Undergraduate Algebraic Geometry.
It has a section on plane curves and proves things in a rigorous way, before going onto things like Hilbert's Nullstellensatz. The book also discusses Affine and Projective varieties before ending on the "27 lines on a Cubic Surface".
I found it very interesting, and it's got some pretty good exercises in it too. The only negative point (in my version at least) is that it wasn't written entirely in LaTeX, but if you can ignore that its a great book to read.