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.