Should I study projective geometry or commutative algebra as prerequisite to start algebraic geometry? I am looking to study Algebraic Geometry but some books list projective geometry as a prerequisite and some list commutative algebra.
I have taken one semester of abstract algebra, real analysis, complex analysis, topology, combinatorics, and differential geometry.
I have not taken a course in projective geometry nor commutative algebra.
Which would be more important as a prerequisite if I want to start learning algebraic geometry?
Do you have any books to recommend?

Marco Flores has given a very good answer. 
I would like to echo one aspect of his answer to suggest that if one wants to learn algebraic geometry, one can just start learning it.  There are several good books available beginning at an undergraduate level, such as Reid's book that Marco Flores mentions.   Later on, one can choose various approaches: the commutative algebra approach of Hartshorne's book, the differential geometry/topology approach of Griffiths and Harris, the related (but more bare-bones) approach of Mumford's book on projective varieties, ... .  There are lots of "road map" questions about algebraic geometry texts here on Math.SE and also on MO that you can consult for guidance.

Finally, let me mention two answers on MO that push against the idea that
algebraic geometry should be regarded as a branch of commutative algebra:
this one and this one.  I agree with their sentiment. 
