Work through this book. While Serre's A Course in Arithmetic (Graduate Texts in Mathematics) is slicker, it is nowhere near as enlightening. Iwaniec's treatise Analytic Number Theory (Colloquium Publications, Vol. 53) (Colloquium Publications (Amer Mathematical Soc)) is a good reference for professionals, but unreadable for someone who has not seen (a lot of) the material before. Davenport's book is very clear and very deep at the same time. The recent editions of this book have been brought up to date, but the core has not changed too much, so don't feel obligated to buy the latest edition.
