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 Hartshrone. I assume you are reading the first chapter.
My advise 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.
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