I am trying to learn about $p$-adic modular forms through N. Katz's text, but I am finding it hard to follow their treatment of classical modular forms in the first chapter. They define an elliptic curve as a map $E \to S$ where the fibers are connected curves of genus 1, with a section $e : S \to E$, and then a modular form is a map which associates to each curve $E/S$ a section of a certain vector bundle $(\omega_{E/S})^{\otimes k}$, satisfying some axioms. They then go on to define level $n$ structure as an isomorphism $\ker n_E \cong (\mathbb{Z}/n\mathbb{Z})^2_S$ and so forth.
It seems like in this book, this introduction is given very much at a level of someone who already knows this -- they give very few proofs and definitions, and tend to gloss over most things. I am looking for a reference which gives a more complete introduction to this perspective on modular forms.