I just wanted to make sure that you're aware that there is another proof of the Mordell conjecture that is in many ways more natural, and that has allowed great generalizations. This is the proof due to Vojta using ideas from Diophantine approximation. Vojta's proof was simplified by Bombieri, and that version does not require very much algebraic geometry, certainly far less than Faltings' original proof. Bombieri's article is quite readable, or you can find the proof with more exposition in my book with Hindry, Diophantine Approximation: An Introduction. Faltings subsequently generalized the methods in Vojta's article to prove strong results concerning rational and integral points on subvarieties of abelian varieties:
(Faltings) Let $A/K$ be an abelian variety defined over a number field. Theorem 1: Let $X\subset A$ be a subvariety. If $X$ contains no translates of abelian subvarieties of $A$, then $X(K)$ is finite. Theorem 2: Let $U$ be an affine open subset of $A$ and let $R\subset K$ be a ring of $S$-integers for some finite set of places $S$. Then $U(R)$ is finite.
This is not to take anything away from Faltings's first proof, which is a tour de force and well worth studying.