I've recently been reading Complex Analysis in One Variable by Narasimhan and Nievergelt. I've done enough work in complex analysis prior that I find the majority of the text quite understandable and beautiful; however, I was pleasantly shocked when I began reading the section entitled "The sheaf of germs of holomorphic functions". After re-reading through the section a number of times, I have found I can formally follow the argument, but lack any intuition for the definitions or the mechanics.
Unfortunately, my passion for analysis has meant the majority of my studies have been directed away from Algebra, and so I've only taken an introductory level course to Group Theory and Ring Theory. Nevertheless, I strongly desire to become more familiar with sheaf theoretic techniques in complex analysis. I could of course simply wait until I have taken more courses in Algebra to become acquainted with it enough to grasp Category Theory at an intuitive level and then begin chipping away at a Sheaf Theory text. However, I am presently only in my first year of college, and I don't wish to be forced to wait a few years until Grad School to appreciate basic applications of Sheaf Theory.
As such, I am wondering if there is a more optimized route to learn the very basic Sheaf Theory I might need to appreciate it's applications to Analysis, and in particular single variable Complex Analysis. I imagine I will not need a complete understanding of modern Sheaf Theory to merely understand its applications to my field. As such, I am wondering what the minimally required topics are that I should study in order to build up some basic intitution for the uses of Sheaf Theory in N&N's text.
Edit: Per the suggestions below, I have checked out copies of Gunning's and Forster's texts on Riemann Surfaces. My initial reaction to both texts has been quite positive; in particular I have found Gunning's text fantastic in that it provides the definition of a sheaf as soon as possible. I look forward to reading through the texts as I find time!
However, I remain completely open to other perspectives on how to best attack the necessary Sheaf theory, and so other answers are welcome!