Modern algebraic geometry vs. classical algebraic geometry

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.

I'm not an algebraic geometer, but I do know several algebraic geometers and it's clear that modern algebraic geometry is a very large field some aspects of which involve technical modern abstractions (stacks!) others of which are in a more combinatorial direction (toric varieties, Grobner bases) and others involve more classical algebraic geometry.

However, I want to remake a point I made on my blog, which is that later on in your career you will be much better at learning things than you are now. As a result it's counterproductive to worry too much about what you should be learning now to maximize your efficiency of learning. Instead you should prioritize things you can learn now and which you enjoy learning now. Certainly you should start with an introductory algebraic geometry book, but once you're done with that there's no harm in looking at Eisenbud and seeing if you enjoy it. But if it's too hard going or if you feel like you're not fully appreciating it then go ahead and try reading something totally different. There'll be plenty of time to learn more commutative algebra while you're a grad student!

I think that some of the answers so far are very good, so this is a bit redundant. I just wanted to emphasize that the distinction between "classical versus modern" algebraic geometry is, to me, not a good one. While it's true that in scheme theory one encounters new phenomena, it can also be used to repair and extend the classical picture. For me, at least, the most beautiful parts of Hartshorne's book are the chapters on curve and surface theory.

As for background, I think that if you read Atiyah-Macdonald and do the exercises, you should be in pretty good shape to get started. I also usually tell my students to learn something about basic manifold theory and algebraic topology, since it provides some useful intuition for a number geometric/homological constructions.