I agree with Donu Arapura's complaint about the artificial distinction between modern and classical algebraic geometry. The only distinction to me seems to be chronological: modern work was done recently, while classical work was done some time ago. However, the questions being studied are (by and large) the same.
As I commented in another post, two of the most important recent results in algebraic geometry are the deformation invariance of plurigenera for varieties of general type, proved by Siu, and the finite generation of the canoncial ring for varieties of general type, proved by Birkar, Cascini, Hacon, and McKernan, and independently by Siu. Both these results would be of just as much interest to the Italians, or to Zariski, as they are to us today. Indeed, they lie squarely on the same axis of research that the Italians, and Zariski, were interested in, namely, the detailed understanding of the birational geometry of varieties.
Furthermore, to understand these results, I don't think that you will particularly need to learn the contents of Eisenbud's book (although by all means do learn them if you enjoy it);
rather, you will need to learn geometry! And by geometry, I don't mean the abstract foundations of sheaves and schemes (although these may play a role), I mean specific geometric constructions (blowing up, deformation theory, linear systems, harmonic representatives of cohomology classes -- i.e. Hodge theory, ... ). To understand Siu's work you will also need to learn the analytic approach to algebraic geometry which is introduced in Griffiths and Harris.
In summary, if you enjoy commutative algebra, by all means learn it, and be confident that it supplies one road into algebraic geometry; but if you are interested in algebraic geometry, it is by no means required that you be an expert in commutative algebra.
The central questions of algebraic geometry are much as they have always been (birational geometry, problems of moduli, deformation theory, ...), they are problems of geometry, not algebra, and there are many available avenues to approach them: algebra, analysis, topology (as in Hirzebruch's book), combinatorics (which plays a big role in some investigations of Gromov--Witten theory, or flag varieties and the Schubert calculus, or ... ), and who knows what others.
It is somewhat jarring to hear of people who "know nothing about the homology theories of topological spaces and their applications" but are "familiar with homological algebra, category theory, spectral sequences (!!)" and the like. Certainly, this is a historically backwards position to be in, since a lot of these highly abstract theories with which you are familiar were almost entirely motivated by the concrete theory which you now wish to master.
Based on what you have said about your background, you will find Peter May's book "A Concise Course in Algebraic Topology" an appropriate read. Peter does not shy away from using categorical or homological machinery when dealing with this material, but also encourages his reader to become adept at the sort of calculations which yield insight into the nature of the subject. Amazingly, you can get the book freely off his website.
Also, to really hammer in the fact that you are engaging with a living, breathing, highly applicable subject, check out Kaczynski, Mischaikow and Mrozek's "Computational Homology" and Edelsbrunner and Harer's "Computational Topology" to find a breadth of applications of homology to physical and life sciences.
Update: The OP and others in a similar position may also be interested in my own upcoming book. You can find the cover here.
Best Answer
(I guess my opinion is no more worthy of being an answer than the opinions in the comments, but it's verbose, so let me put it in the answer box anyway.)
As a beginning PhD student I knew a reasonable amount of algebraic topology, similar to what you describe in the question. But I don't think it really gave me any extra or better choices in how to start learning algebraic geometry. As Steven Landsburg's comment suggests, the kind of objects one studies and the methods one uses in algebraic geometry are so much more specialised than arbitrary (even nice) topological spaces that you really need to start from scratch, with something like Shafarevich (as suggested by Mark Grant).
That isn't to say that having a good knowledge of algebraic topology won't be useful to you in learning algebraic geometry --- quite the opposite, in fact. (Example: characteristic classes.) And, as you progress in algebraic geometry, you will likely run into more and more topics where your topology knowledge gives you a great head-start on understanding. But it won't really help with those first steps.
On the other hand, if I had to nominate a beginning algebraic geometry textbook that is oriented towards the topological point of view, I guess it would be Principles of Algebraic Geometry by Griffiths and Harris. But, great resource though it is, I don't think I can recommend that book to anyone in good conscience as a first introduction to algebraic geometry.