There are the lecture notes "AN INTRODUCTION TO AUTOMORPHIC REPRESENTATIONS" by Jayce R. Getz, which start on the background on adele rings, then treat algebraic groups and automorphic representations, Nonarchimedian Hecke algebras, a bit of archimedian representation theory,
before in chapter $6$ it comes to automorphic forms on adele groups. This seems quite appropriate. For the cuspidal spectrum see [Don82] of the references, too.
The link is as follows: http://www.math.duke.edu/~jgetz/aut_reps.pdf
To address the stated subtext of your post: many people, myself included, take the position that groups are important because they act on things. A representation is just a group action on a vector space (by linear operators). And whenever you have a group action, even if it isn't on a vector space, there is often a closely related representation lurking nearby.
For example, if $G$ is acting on a finite set $S$ then there is an induced representation on $\mathbb{F}^S$ for any field $\mathbb{F}$. More generally, if $G$ is acting on a space $X$ then there is an induced representation of $G$ on the algebra $\mathcal{O}(X)$ of functions on $X$, whatever we mean by "function" in this situation (e.g. perhaps $X$ is a topological space on which $G$ acts by homeomorphisms, and $\mathcal{O}(X)$ is the algebra of continuous real-valued functions on $X$).
To elaborate on this last point: suppose $X$ is a set on which $G$ acts. For any set $Y$, there is an induced action on the set $Y^X$ of maps from $X$ to $Y$: for $g\in G$ and $f:X\to Y$, the map $g\cdot f$ is defined by
$$(g\cdot f)(x) = f(g^{-1}x).$$
If we equip $X$ with some additional structure, like a topology, we may insist that the action of $G$ preserve this structure. This guarantees that the action of $G$ on maps will take structure-preserving maps to structure-preserving maps. As an example, if $X$ and $Y$ are topological spaces and $G$ acts on $X$ by homeomorphisms, then the action of $G$ on $Y^X$ sends continuous maps to continuous maps. In particular, taking $Y=\mathbb{R}$, we have an action of $G$ on the algebra of real-valued continuous functions.
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In Bruce Sagan's book, $The$ $Symmetric$ $Group$ (second edition), he proves a unimodality theorem using representation theory---if I recall correctly, for the poset called $L(m,n)$---but this wasn't the first proof of that theorem (Corollary 5.4.10).