What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

Here are two references which seem to answer your request:
I) Lectures on Curves, Surfaces and Projective Varieties by Beltrametti, Carletti, Gallarati, Bragadin.
This is a fat textbook written by four Italian geometers in a very classical style and concentrating on classical projective geometry: schemes, cohomology or functors are never even alluded to!

II) Geometria proiettiva, Problemi risolti e richiami di teoria by Fortuna, Frigerio, Pardini.
This is a book consisting of solved exercises preceded by a reminder of the theory.
Although the book is recent the content is very classical and elementary: cross-ratios, quadrics, pencils of conics, inflection points,linear systems,...
From a review:
“This book is the result of the experience acquired by the authors while lecturing Projective Geometry to students from a three year course leading to a degree in Mathematics in the University of Pisa (Italy). … ” (Ana Pereira do Vale, Zentralblatt MATH, Vol. 1227, 2012)
(The book is in Italian, but judging from your first name this might not be a big deal :-)
Anyway mathematical Italian is very close to mathematical English)

I think there is some poetic justice in the fact that all seven authors of the two books are Italian: the justly vaunted Italian algebraic geometry seems to be alive and well in its native country!

Edit
Richter-Gebert has has recently written an encyclopaedic book containing an amazing wealth of material on projective geometry, starting with nine (!) proofs of Pappos's theorem .
The book examines some very unexpected topics like the use of tensor calculus in projective geometry, building on research by computer scientist Jim Blinn .
It would be difficult to read that book from cover to cover but the book is fascinating and has splendid illustrations in color.

My recommendations are:

[1] Lynn E. Garner: An Outline of Projective Geometry

[2] A. Seidenberg: Lectures in Projective Geometry

[3] Robin Hartshorne: Foundations of Projective Geometry, http://filebox.vt.edu/users/jabrunso/Math/Hartshorne.pdf

All these books are on classical projective geometry, assuming only basic knowledge. But I think your familiarity with algebraic geometry gives you a higher point of view while reading any of these books. (My personal opinion is that [1] is the best, because it covers the most impotant synthetic results - Desarguesian and Pappian projective planes, projectivities, collineations, polarities, conics, etc. - and the linear algebraic point of view as well.)

Here are some recommendations from my reading list:

1. Linear Geometry by Gruenberg and Weir

2. Projective and Polar Spaces by Peter Cameron

3. Projective Geometry: From Foundations to Applications by Albrecht Beutelspacher and Ute Rosenbaum

4. Outline of Projective Geometry by L. E. Garner