Introductive Book on Modular Forms

The first book you want to look at is Serre's "A course in arithmetic". THe second is the aptly named "A first course in modular forms"

Step $0$: J W Brown & R V Churchill - Complex Variables and Applications [→];

Step $\frac12$: E Freitag & R Busam : Complex Analysis [→];

Step $1$: E Freitag - Complex Analysis 2 [→].

The book---"A first course in modular forms" by F. Diamond, J. Shurman is a good book to start to study classical modular forms. The advanced one--- "Modular forms" by Toshitsune Miyake is also a very good textbook to learn modular forms. Good luck.