I recommend you a book that I thoroughly enjoyed when I studied complex analysis:
"An Introduction to Complex Analysis", by Agarwal, Perera and Pinelas.
This book is divided into 50 lectures that will take you from the very basics of complex analysis to advanced topics; it starts with the usual introductory material and then goes through the Cauchy integral formula, power series, the residue theorem and evaluation of real integrals by contour integration, conformal mappings, and harmonic functions. It ends with brief sections on the Riemann zeta function, Riemann surfaces, and the Bieberbach conjecture. Almost every theorem/lemma/proposition is proved in a rigorous way.
At the end of each lecture there are numerous exercises to test your knowledge; hints are provided for most of them so, if you're studying on your own, you won't be kept in the dark forever wondering how to approach a specific problem.
If you're lucky enough, perhaps you'll even be able to access the digital version of the book for free through your university's library platform, at SpringerLink.