The books by Silverman can't be beat and I won't simply repeat what's been said below.

Adding to the books already mentioned, Lawrence Washington's recent text is supposed to be excellent, but I haven't seen it yet.

Miles Ried also wrote a beautiful set of lecture notes that was used at Cambridge for many years-they may or may not still be available online for download.

And of course no introduction to algebraic geometry through elliptic curves would be complete without mentioning the classic introduction to algebraic curves by William Fulton, which is available online free for download by googling it.

The books by Silverman can't be beat and I won't simply repeat what's been said below.

Adding to the books already mentioned, Lawrence Washington's recent text is supposed to be excellent, but I haven't seen it yet.

Miles Reid also wrote a beautiful set of lecture notes that was used at Cambridge for many years-they may or may not still be available online for download.

And of course no introduction to algebraic geometry through elliptic curves would be complete without mentioning the classic introduction to algebraic curves by William Fulton, which is available online free for download by googling it.