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.
This is by no means a comprehensive answer, but I'll risk some remarks. Briefly, my impression is that topology often tells one what to expect, but does not always tell how to prove it. In case it matters, this is an impression of someone whose first and true love is geometric topology, but who is interested in algebraic geometry as well.
There are some topological notions that have analogs in algebraic geometry. The best known is perhaps the \'etale cohomology. It has some properties very similar to the "topological" cohomology, i.e. the cohomology of constant or more generally, constructible sheaves. There is the Mayer-Vietoris sequence (for a Zariski open cover); furthermore one can define \'etale constructible sheaves, which gives the relative cohomology of a couple (a variety, a closed subvariety). One can define the constructible derived category, and there are the "six operations": the direct and inverse image, the direct and inverse image with compact support, RHom and the derived tensor product. Moreover, there is the Verdier duality (and hence, the Poincar\'e duality as well). There is the cohomology class of a cycle and so one can define the Chern classes of a vector bundle.
There are ways to compare the \'etale cohomology and the topological cohomology. For example, let $k$ be an algebraically closed field of finite characteristic. Then we can apply the Witt vector procedure http://eom.springer.de/W/w098100.htm to it to get a complete discrete valuation ring with residue field $k$ and fraction field of characteristic 0. Then, if we have a smooth scheme over $R$, we can apply the procedure explained in SGA 4 1/2, p.54-56 to construct a morhism from the cohomology of the fiber over the maximal ideal of $R$ to the (\'etale) cohomology of the fiber over the algebraic closure of the fraction field. (And see pp. 52-53 there for an analogy with the cohomology of the preimage of a disk under a holomorphic mapping and the preimage of the origin.) Then one can use M. Artin's comparison theorem to construct an isomorphism with the usual "topological" cohomology of the constant sheaf. The resulting maps are not isomorphisms in general but they are functorial with respect to maps of smooth varieties over $R$.
Perhaps, the \'etale cohomology smooth complete varieties is a bit too close to the cohomology of complex algebraic varieties. For example, the \'etale cohomology of the projective line over an algebraically closed field with coefficients in a finite abelian group $A$ of order prime to the characteristic of the field is $A$ in degrees 0 and 2 and 0 elsewhere, just as in the complex case. But in the complex case this is ultimately a consequence of the fact that $\mathbf{C}$ is 2-dimensional over $\mathbf{R}$. So why do fields of positive characteristic know about it? To me this is a bit mysterious.
Here is a somewhat less trivial example. Morse theory gives a CW complex homotopy equivalent to a given manifold once we have a strict Morse function on the manifold. As indicated in the paper http://arxiv.org/abs/math/0301140 by D. Arapura, the algebraic analog of a cell is probably an affine variety $X$ and a constructible sheaf on it whose cohomology vanishes in degrees other than $\dim X$. Given a quasiprojective $X$ we can construct a cell decomposition (of sorts). First we replace $X$ with an affine $Y\to X$ such that the fiber over any closed point is an affine space. This is the Jouanolou trick and a proof of its existence is sketched e.g. here The Jouanolou trick. Then we can take any constructible sheaf $F$ on $X$ and pull it back to $Y$. Then we use Beilinson's lemma to choose a closed subvariety $Y'\subset Y$ such that $H^*(Y,Y',F)=0$ except maybe in degree $\dim Y$ (the existence of such a $Y'$ can be proven using the usual Morse theory if one is working over $\mathbf{C}$). Then we apply the same procedure to $Y'$ and so on. We get a filtration of $Y$ whose Leray spectral sequence will be concentrated in the 0-row. This is an analog of the cellular complex.
Since this is already way too long, let me briefly mention the differences between the algebraic and the topological cases, the way I understand them. First, there are some tools in topology that have no analog in algebraic geometry. For example, everything involving partitions of unity is a no-no. In fact I don't know any example of the use of fine sheaves in algebraic geometry. So while there is an analog of Sard's theorem, some of its consequences fail miserably. For example, there are smooth complete complex varieties that can't be embedded in any projective space. (These examples, due to Hironaka, are described e.g. in Hartshorne, Appendix B.) On the other hand, in finite characteristic there is the Frobenius automorphism which acts on everything. For complex algebraic varieties there is one of the consequences, the weight filtration, but there is no Frobenius so the proof of its existence is a bit roundabout.
Best Answer
Do you consider $L$-functions of elliptic curves over $\mathbf Q$ (or other number fields) to be in the spirit of "analytic number theory undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on"? Those 19th and early 20th century folks did not have the definition, which only came much later in the 20th century, but the idea of defining such functions as an Euler product and then Dirichlet series, and seeking an analytic continuation and functional equation, is a task they would have understood. Deuring proved the analytic continuation and functional equation in a special case (CM elliptic curves) in the 1950s, but the case of all elliptic curves over $\mathbf Q$ was settled using ideas coming from the proof of Fermat's Last Theorem, hence using modern algebraic geometry.
The Sato-Tate conjecture is an analytic conjecture somewhat in the spirit of the prime number theorem. It was formulated in the 2nd half of the 20th century but could have been appreciated earlier. Like the prime number theorem, which is equivalent to nonvanishing of the zeta-function on the line ${\rm Re}(s) = 1$, the Sato-Tate conjecture was known to be a consequence of analyticity and nonvanishing of certain $L$-functions on vertical lines (boundary of right half-planes) and those $L$-function properties were proved about 10 years ago with algebro-geometric methods.