I want to grasp the theory of microlocal analysis and apply this theory to some PDEs in $\mathbb{R}^n$. But most textbooks I found put much priority on manifolds. Sadly, I know little about them and don't really care about them. Also, I find that their analysis are beautifully but quantitative. I'm a man of classical PDEs and like estimates a lot. On the other hand, I'm also familiar with linear/nonlinear functional analysis. To sum up, I hope the textbooks:
1: pay a lot of attention to PDEs in $\mathbb{R}^n$ or bounded domains.
2: from the view points of functional analysis.
There are many good resources (although they aren't necessarily the easiest to read!).
Hormander's first book in his sequence of $4$ books on linear partial differential operators hits on some core concepts in microlocal analysis at the end in the form of wavefront sets and propagation of singularities (if you want to see more PDE, then check out all $4$) and covers pseudodifferential operators in book $3$. You could also check out Microlocal Analysis for Differential Operators by Grigis and Sjostrand, Michael Taylor's book on pseudodifferential operators (much of the standard material is also covered in chapter 7 of Taylor's second PDE volume), or Melrose's Introduction to Microlocal Analysis. If you don't mind a semiclassical approach (introduction of parameter $h$, which you consider in the semiclassical limit $h\rightarrow 0$), which is quite popular, then you could look at Zworski's Semiclassical Analysis or Martinez's An Introduction to Semiclassical and Microlocal Analysis.
All of the listed books spend plenty of time talking about Euclidean space (with some touching on manifolds later on).
EDIT: Oh, and you might also like to take a look at Folland's Harmonic Analysis in Phase Space, chapters $2$ and $3$ in particular.