I am soon to become a graduate student and so I started a personal project; I want to understand Faltings's proof of the Mordell conjecture.

I want to get into arithmetic geometry (since I always liked both algebraic geometry and number theory) and I thought that understanding this proof might be a good start (mostly because the first time a read about it I was quiet surprised and intrigued).

I am already trying to formalize my knowledge on modern algebraic geometry studying from Hartshorne's "Algebraic Geometry" and Liu's "Algebraic Geometry and Arithmetic Curves".

So, my question is where should I continue after I "finish" formalizing my knowledge on modern algebraic geometry?

I know there's a book called "Arithmetic Geometry" edited by Silverman and Cornell which actually contains Faltings's proof, but I am not sure if the material covered in Liu's and Hartshorne's books is enough to dive into this more specialized content.

I'm also aware of Milne's notes on Abelian Varieties which contains Faltings's proof, but I haven't look at these in detail yet.