I'm an undergraduate student of mathematics, but soon I'll graduate, and as a personal project I want to understand Falting's Theorem, specifically I want to understand Falting's proof; but yet I have no clue where to start studying.
I already have some notions on Scheme Theory and I have studied classical algebraic geometry before. So my first guess is that I need to consolidate my knowledge on modern algebraic geometry and then move to it's applications to curves.
I made some research before asking this question and I found "Algebraic Geometry and Arithmetic Curves" by Qing Liu, I guess that might be a good start, he even states in the preface that the material covered in the book will allow the student to approach more specialized books like "Arithmetic Geometry" edited by Silverman and Cornell (this books contains the proof of Falting's Theorem).
I also found these notes of a Stanford seminar about the topic, which might help.
Finally I found Milne's notes on Abelian Varieties in which he gives the proof of Falting's theorem.
So, my question is simple, where do I start and which book or notes should I follow?