I have a copy of Mumford's book, but as a final year undergraduate I am finding it to be a little too dense as a starting text. Something lighter would be appreciated to get an intuition before approaching Mumford and trying to tackle Mordell-Weil.
However, besides this book, every other reference I can find seems to concern itself with the analytic theory.
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There is a FANTASTIC book currently being written by Ben Moonen et. al. See here. If you are very algebro-geometrically minded, this should be perfect for you.
I think a good reference are the lecture notes of J.S. Milne on Abelian Varieties. The first chapter treats a lot of basic algebraic theory.