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?
On
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.
I think Commutativa Algebra is more important, yet there are very good books (e.g. Klaus Hulek's Elementary Algebraic Geometry, Fulton's Algebraic Curves) that introduce the necessary concepts and results from commutative algebra as they are needed. So If you don't really care about Commutative Algebra itself (or its applications in other areas like algebraic number theory), I suggest you start already with introductory algebraic geometry (Reid's Undergraduate Algebraic Geometry is another good reference).
Projective geometry is something you will definitely learn along the way.