Roadmap for studying arithmetic geometry

My suggestion, if you have really worked through most of Hartshorne, is to begin reading papers, referring to other books as you need them.

One place to start is Mazur's "Eisenstein Ideal" paper. The suggestion of Cornell--Silverman is also good. (This gives essentially the complete proof, due to Faltings, of the Tate conjecture for abelian varieties over number fields, and of the Mordell conjecture.) You might also want to look at Tate's original paper on the Tate conjecture for abelian varieties over finite fields, which is a masterpiece.

Another possibility is to learn etale cohomology (which you will have to learn in some form or other if you want to do research in arithemtic geometry). For this, my suggestion is to try to work through Deligne's first Weil conjectures paper (in which he proves the Riemann hypothesis), referring to textbooks on etale cohomology as you need them.

If you can find a (say, library) copy of Cornell and Silverman's Arithmetic Geometry I would highly recommend it. It is a comprehensive treatment of the arithmetic theory of abelian varieties using the modern scheme-theoretic language. Lamentably it's basically impossible to buy a copy these days (there's usually one available on-line from some obscure seller for something like $950). I also agree with the above recommendations of Liu's Algebraic Geometry and Arithmetic Curves. It builds scheme theory from scratch (even developing the necessary commutative algebra in first chapter) and has an eye towards arithmetic applications throughout. In particular, the end of the book has a great chapter on reduction of curves. If you want a treatment of elliptic curves in extreme generality (using scheme language) then you might be interested in Katz' and Mazur's Arithmetic Moduli of Elliptic Curves. I emphasize however, that this particular book is very difficult (at least for me it is).

An apology first: This is more a supplement to Charles' answer than an answer itself. This was originally a set of comments, but I was not able to format the comments so as to be readable.

"Arithmetic of Elliptic curves" is particularly recommended for those who want a first look at arithmetic applications of cohomology. Chapter 8 proves the Mordell-Weil theorem using Galois cohomology. Pretty much everything in this book is good though and the only overlap with Hartshorne is in the first two chapters. It's the canonical book for elliptic curves for a reason!

"Rational Points on Elliptic curves" would probably not be so exciting for someone who's already gone through Hartshorne.

"Advanced Topics" is exactly that, but maybe a little more friendly than most topics books. The chapters are essentially free standing. Of particular interest might be the chapter on Elliptic surfaces which give a peek at ℤ schemes in (almost) all their glory.

I've only glanced through Hindry-Silverman, so I couldn't say much either way.

"An Invitation to Arithmetic Geometry" for this reader would primarily serve to highlight how Algebraic Number Theory intersects Arithmetic Geometry, I think.

"Algebraic Geometry and Arithmetic Curves" is a fantastic reference for Arithmetic Geometry, and there's quite a lot of overlap with Hartshorne.

edit: For moduli of elliptic curves, Chapter 1 (Modular forms) of "Advanced topics" is a good place to start, and Katz-Mazur is a good eventual target. Between those two, there are lots of books on modular forms and moduli spaces to fill the gap. I'm partial to Diamond and Shurman, but the original works of Shimura deserve recognition here. Your mileage may vary.