In terms of analyzing sums of the form
$$
S(x) = \sum_{n\leq x} f(n),
$$
where $f$ is some number-theoretic function, I don't think there's one unified approach for tackling all sums of this kind. As such, I think it's more useful to describe/recommend some general methods rather than books on the subject of asymptotic analysis.
Since you've had a course in analytic number theory, I assume you're familiar with partial summation (also called Abel summation) and Dirichlet's hyperbola method. For the sums $S(x)$ that arise in analytic number theory, these should be the first things you look at.
If $f$ is a multiplicative function, then Perron's formula is a good starting place. This will require knowledge of the associated Dirichlet series
$$
\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \prod_{p}\left(1+\frac{f(p)}{p^s}+\frac{f(p^2)}{p^{2s}}+\cdots \right)
$$
and its behavior as a complex function (the fact that one is able to express the Dirichlet series on the left as an Euler product on the right is equivalent to $f$ being multiplicative). If $f$ is can be understood well enough, one can often express the Dirichlet series in terms of familiar $L$-functions, where information such as poles, Laurent series expansions, analytic/meromorphic continuations, and functional equations are known. By shifting contours, one can express the value of $S(x)$ as the residue (or sum of residues) at special points of $L$-functions (or derivatives and/or products thereof) plus error terms involving integrals of $L$-functions.
Sometimes there are summation formulas that one can use to study such sums in place of the theory of complex-valued $L$-functions. For an introduction of this sort, I would recommend Chapter 4 of Analytic Number Theory by Iwaniec and Kowalski (if you're seriously interested in analytic number theory, this book is worth the investment. I refer to my copy on a weekly if not daily basis). Very often, the determination of the main term of the asymptotic is the easy part; in multiplicative number theory, it is often a straightforward (though potentially tedious) residue computation.
The meat of the analysis is usually the estimation of the error term, and this is where tools like Poisson/Voronoi summation (see also Chapter 10 of Cohen's Number Theory, Volume 2 - Analytic and Modern tools), statistics of $L$-functions (such as subconvexity and moment estimates), and even the deep subject of spectral theory become relevant (the classic but challenging text on spectral theory is Iwaniec's Spectral Methods of Automorphic Forms).
General introductions such as Apostol's Introduction to Analytic Number Theory and its sequel Modular Functions and Dirichlet Series in Number Theory are good starting places to understanding averages of arithmetic functions such as $S(x)$. Part of the fun of number theory is that, although general methods like Perron's formula and Poisson summation can be useful, many specific problems require ad hoc ideas that are specific to the problem at hand; the best error terms are usually obtained by taking advantage of the specific structure/properties inherent in the functions being studied.
Hopefully that's helpful to you. Often times, I find its more useful to try to look up references on specific problems, rather than consulting texts on general methods. The literature on specific problems will usually include references (either formal references or buzz words) to general or specific methods used in tackling the problem, and these are the things you can then track down and learn more about. If your main interest is in sums involving the divisor function, you should do a deep dive into Dirichlet's hyperbola method and Voronoi summation, as these are absolutely essential tools in studying such sums.
