Currently, I am organizing a potential reading course on Elliptic Curves and have gotten some reference suggestions (I will be a student participating in said course), but do not have a really clear idea of what one might hope to cover or gain from it. For motivation, I am interested in elliptic curves due to potentially seeing Algebraic Geometry in action (and I am somewhat interested in Algebraic Number Theory, but am only taking a first graduate course in the material this semester).
I have gotten some reference suggestions such as: "Introduction to Elliptic Curves and Modular Forms" by Koblitz, the text by Silverman and Tate, and Silverman's "The Arithmetic of Elliptic Curves". There were also the notes by Milne.
After an inspection of Silverman's "The Arithmetic of Elliptic Curves", Chapters V on Elliptic Curves over Finite Fields and Chapter VIII with the Mordell-Weil Theorem for $\mathbb{Q}$ shine to me. However, having such a narrow goal like proving the Mordell-Weil Theorem means I will inevitably omit many other sections.
What are topics in Elliptic Curves that are essential to be aware of and should be covered in a reading course? Potentially, this may include material outside of Silverman's text.