What are the recommended books for an introductory study of elliptic curves?

Silverman and Tate to start, then Silverman, and finally Silverman again. These are basically canonical references for the subject.

The other book suggestions are all so far excellent; the only caveat with them is that they all get into the number theoretic aspects very soon. I am taking the guess that you are more geometrically minded since you are starting with algebraic geometry rather than with number theory.

Also I am taking the guess that you are reading algebraic geometry from the standard book of Hartshorne. I assume you are reading the first chapter.

My advice to you would be to first understand affine and projective varieties as given in chap I of Hartshorne, and then move straight ahead to chapter IV on algebraic curves. You would have to take a few things like the Riemann-Roch theorem (rather, Serre duality theorem) for granted and you would have to replace any occurrence of "scheme" with variety, and there may be a few gaps. I suggest that you ignore these and read it. This will give you a very solid and rather modern introduction into the subject algebraic curves, and to elliptic curves in particular.

Afterwards you can go back to chaps. II and III and read the theory of schemes and the machinery of sheaf cohomology, if you wish to further pursue algebraic geometry.

I highly recommend Elliptic Curves by Alain Robert. It is very clearly written, has few prerequisites, yet brings the reader straight into the connection between the complex analytic side of algebraic curves and the algebro-geometric side. It eventually discusses $p$-adic curves and their relation to $p$-adic analytic functions, as well as using these to prove the main theorem of complex multiplication (that $j(\tau)$ is an algebraic integer).