I am going to take a course in Algebraic Geometry covering chapter II and III of Hartshorne next year. Before this, I have the option to take either Commutative Algebra (Mainly use Atiyah) or Riemann Surface & Algebraic Curves (Mainly Rick Miranda) next semester.
As I know, commutative algebra provides the necessary algebraic tools to handle algebraic geometry. Some said it can be learnt along Hartshorne while some said it is better to learn commutative algebra before learning Hartshorne.
Also, some mentioned Riemann Surface & Algebraic Curves (over $\mathbb{C}$) provides geometrical intuition as they are the prototype to modern algebraic geometry which helps in understanding the concept in Hartshorne.
Which is more important for me to move into Chapter II and III of Hartshorne? I wish to pick both but unfortunately, I am overload next semester so I can only pick one.
By the way, I had taken a course in algebraic curves where lecture note by William Fulton was used. Thus, I know a bit commutative algebra from there but not as much as in Atiyahs'
To understand chapters II-III of Hartshorne you need much of the material in Atiyah-Macdonald. The material in Miranda's book will help a lot with the motivation, however while the definitions and results in Hartshorne may sometimes resemble those in Miranda's book, the arguments will almost always be done using commutative algebra. As such, to understand Hartshorne you should take commutative algebra.
Ideally, of course, you would take both classes. However, since you already had a course in algebraic curves it shouldn't be a serious issue to not take the course on complex curves.