Is there a "unifying framework" for harmonic analysis? Recently, I was exposed to a basic harmonic analysis course.
Although the course is almost over, I still can't put my finger on what harmonic analysis is about. I have a vague idea that it is supposed to be able to somehow provide a (more or less) nice description of (representatives of iso classes of) irrep's of topological groups (or maybe just in special cases, e.g. $S^1$ in the case of the "classical" harmonic analysis...?).
Adopting this rather primitive point of view, the theory further developed seemed to me as a case-by-case treatment of (classes of) individual examples.
To be more specific, in the first lecture I was presented with a list of "algebraic/geometric objects" and the corresponding "spectral objects". The list seemed a bit random to me (in the sense that I could not find any really strong similarities between the algebraic object-corresponding spectral object relations). So the first part of this question is

is there some general description what a spectral object (in the context of harmonic analysis) is?

Although this question certainly seems as a soft question, it is meant more as a reference request - in particular,

are there any texts (preferably textbooks or lecture notes) with an
  emphasis on some conceptual tratment of harmonic analysis?

Although I would appreciate a reference to some text, some clarification of the concept of harmonic analysis are also welcome.
Thanks in advance for any help.
 A: Harmonic analysis means different things to different people, but to me it's describing the decomposition into irreducibles of the regular representation of a group $G$ on the space $L^2(G)$ of square-integrable $f:G\rightarrow\Bbb{C}$ via some appropriate "Plancherel theorem''.
This may be something you're familiar with, but I'll say it anyway: the idea is that Fourier series (in the classical sense) give a way of writing a function on $S^1$ in terms of a sum over $e^{inx}$ with coefficients, and $x\mapsto e^{inx}$ are precisely the set of unitary characters of $S^1$. This generalises very nicely to arbitrary compact groups via the Peter-Weyl theorem. This is often stated in various, essentially equivalent ways, but let's just treat it as saying that the regular representation of $G$ on $L^2(G)$ decomposes as a direct sum of representatives of the irreducible unitary representations of $G$, with multiplicity.
That's one half of the main question in Harmonic analysis -- the other is how some specific function on $G$ decomposes with respect to this, which is (more or less) to say that if you form the $G$-orbit of some $L^2$ function under the regular representation, which unitary representations do we see? This is dealt with through Plancherel formulae and "Fourier series", and once one moves beyond compact groups through computations of Plancherel measures (but I'm not really qualified to say anything too detailed about that).
As for "geometric" and "spectral" objects, that's something that I'm less sure is really well-defined. One nice way to explain what this means in a general (i.e. not specialized to harmonic analysis) setting is the (Kac's?) question of whether one can "hear the shape of a drum". This question should be interpreted as taking some geometric object (the shape of a drum), and seeing what it "does" (what sound it makes when you hit it). Something that it "does" is a spectral object. This (sort of) fits in with the way that I'm familiar with the word "spectral" being used in this context, in that you have your object (a group, let's stick with compact for simplicity), and something that it "does" (acts on the Hilbert space $L^2$). Then harmonic analysis gives a way of decomposing this action, in a way that I suppose is somewhat analogous to decomposing an operator on a Hilbert space into eigenspaces.
There's another way that the terms "geometric" and "spectral" turn up in harmonic analysis, and that's when you're using trace formulae. Here, I'm really unqualified to comment, but one has a "geometric" side of the formula, which is a sum over various orbits of an action by conjugacy and "orbital integrals" defined in terms of some $L^2$ function. This is then equal to a "spectral" side, which is a sum over irreducible unitary representations of traces of the representation.
I'm sorry that this has all been a bit vague, but I'm far from being an expert on harmonic analysis -- it's just an area that proves interesting results in my field in a way that I don't understand! I don't really know of a single reference that explains all of this nicely; I got most of my understanding by bouncing between various references and playing around with examples. Off the top of my head, I remember liking Terry Tao's notes on Peter-Weyl, Deitmar and Echterhoff's Principles of Harmonic Analysis (which apparently also has a more gentle precursor which handles the less abstract side of things, although I've never looked at it), and Folland's Course in Abstract Harmonic Analysis. On the more number theoretic side, there's also Ramakrishnan and Valenza's Fourier Analysis in Number Fields. I'd definitely recommend doing a few simple examples where you should be able to work out the theory completely. Maybe $\Bbb{Z}/n\Bbb{Z}$, $S_n$ for some small $n\geq 3$, and some nice, non-trivial compact Lie group or something along those lines (the key to choosing a good instructional example being in choosing one where you can just forget about topologies because it's "obvious" which representations you're working with, and then just read off from a character table / classification of irreducibles).
