Speaking completely from personal experience, I learned almost everything I know in terms of asymptotics and bounding integrals from doing analytic number theory problems.
These specific types of manipulations didn't actually come up in any of my analysis courses over the years. However, knowing basic analysis and how to work with limits is very important, and is a definite prerequisite. Rudin's "Principles of Mathematical Analysis 3E" is the standard text for an introduction to analysis. Understanding the basics in the first 1-7 chapters and having some basic complex analysis is fundamental.
If you are already familiar with these concepts, then I suggest working through the proofs and the exercises in your analytic number theory book.
Remark: Both of the books you mention are excellent, but I would not usually recommended them if you have never seen analytic number theory before.
I personally used Montgomery and Vaughn's book, which I quite like, but I have heard very good things about Davenport's Multiplicative Number Theory. This Math Stack Exchange question deals with books for analytic number theory.