I believe there have been similar questions, but not one exactly of this flavor.
To answer your last question, it is true that you need to know many different areas of mathematics in order to delve deeply into algebraic geometry. On the other hand, to get a basic grounding in the field, one need only have a basic understanding of abstract algebra.
That being said, I will give my recommendations.
If you have already done complex variables, and I'm not sure that every student in your position will have completed this, I recommend Algebraic Curves and Riemann Surfaces by Rick Miranda. Although this book also develops a complex analytic point of view, it also develops the basics of the theory of algebraic curves, as well as eventually reaching the theory of sheaf cohomology. Multiple graduate students have informed me that this book helped them greatly when reading Hartshorne later on.
If you want a very elementary book, you should go with Miles Reid's Undergraduate Algebraic Geometry. This book, as its title indicates, has very few prerequisites and develops the necessary commutative algebra as it goes along. More advanced students may complain that this book does not get very far, but I think it may very well satisfy what you are looking for.
Another book you might want to check out is the book Algebraic Curves by William Fulton, which you can thankfully find online for free.
If you would not mind a computational approach, and furthermore a book which requires even fewer algebraic prerequisites than you seem to have, you might want to check out Ideals, Varieties, and Algorithms by Cox and O'Shea.
Thierry Zell's suggestion is also supposed to be good.
That being said, if you decide that you like algebraic geometry and decide to go more deeply into the subject, I highly recommend that you learn some commutative algebra (such as through Commutative Algebra by Atiyah and Macdonald). But for the moment, I think the above recommendations will suit you well.
Best Answer
Here are a few books on the history of recent mathematics that I recommend:
A History of Algebraic and Differential Topology, 1900 - 1960 by Jean Dieudonne.
History of Topology by I.M. James.
Reciprocity Laws: From Euler to Eisenstein by Franz Lemmermeyer.
The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae
by Catherine Goldstein, Norbert Schappacher, and Joachim Schwermer.
The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators by Alexander Soifer.