The book you are looking for exists!! And indeed it contains ALL the buzzwords in your question!
It is Manin/Panchishkin's "Introduction to Modern Number Theory". This is a survey book that starts with no prerequisites, contains very few proofs, but nicely explains the statements of central theorems and the notions occurring therein and gives motivations for the questions that are being pursued. You should take a look, at least it can help you decide what you want to study in more detail.
Caveat: in order to give you an overview, I've been vague/sloppy in several places.
Well the basic link to representation theory is that modular forms (and automorphic forms) can be viewed as functions in representation spaces of reductive groups. What I mean is the following: take for example a modular form, i.e. a function $f$ on the upper-half plane satisfying certain conditions. Since the upper-half plane is a quotient of $G=\mathrm{GL}(2,\mathbf{R})$, you can pull $f$ back to a function on $G$ (technically you massage it a bit, but this is the main idea) which will be invariant under a discrete subgroup $\Gamma$. Functions that look like this are called automorphic forms on $G$. The space all automorphic forms on $G$ is a representation of $G$ (via the right regular represenation, i.e. $(gf)(x)=f(xg)$). Basically, any irreducible subrepresentation of the space of automorphic forms is what is called an automorphic representation of $G$. So, modular forms can be viewed as certain vectors in certain (generally infinite-dimensional) representations of $G$. In this context, one can define the Hecke algebra of $G$ as the complex-valued $C^\infty$ functions on $G$ with compact support viewed as a ring under convolution. This is a substitute for the group ring that occurs in the representation theory of finite groups, i.e. the (possibly infinite-dimensional) group representations of $G$ should correspond to the (possibly infinite-dimensional) algebra representations of its Hecke algebra. This type of stuff is the basic connection of modular forms to representation theory and it goes back at least to Gelfand–Graev–Piatestkii-Shapiro's Representation theory and automorphic functions. You can replace $G$ with a general reductive group.
To get to more advanced stuff, you need to start viewing modular forms not just as functions on $\mathrm{GL}(2,\mathbf{R})$ but rather on $\mathrm{GL}(2,\mathbf{A})$, where $\mathbf{A}$ are the adeles of $\mathbf{Q}$. This is a "restricted direct product" of $\mathrm{GL}(2,\mathbf{R})$ and $\mathrm{GL}(2,\mathbf{Q}_p)$ for all primes $p$. Again you can define a Hecke algebra. It will break up into a "restricted tensor product" of the local Hecke algebras as $H=\otimes_v^\prime H_v$ where $v$ runs over all primes $p$ and $\infty$ ($\infty$ is the infinite prime and corresponds to $\mathbf{R}$). For a prime $p$, $H_p$ is the space of locally constant compact support complex-valued functions on the double-coset space $K\backslash\mathrm{GL}(2,\mathbf{Q}_p)/K$ where $K$ is the maximal compact subgroup $\mathrm{GL}(2,\mathbf{Z}_p)$. If you take something like the characteristic function of the double coset $KA_pK$ where $A_p$ is the matrix with $p$ and $1$ down the diagonal, and look at how to acts on a modular form you'll see that this is the Hecke operator $T_p$.
Then there's the connection with number theory. This is mostly encompassed under the phrase "Langlands program" and is a significantly more complicated beast than the above stuff. At least part of this started with Langlands classification of the admissible representation of real reductive groups. He noticed that he could phrase the parametrization of the admissible representations say of $\mathrm{GL}(n,\mathbf{R})$ in a way that made sense for $\mathrm{GL}(n,\mathbf{Q}_p)$. This sets up a (conjectural, though known now for $\mathrm{GL}(n)$) correspondence between admissible representations of $\mathrm{GL}(n,\mathbf{Q}_p)$ and certain $n$-dimensional representations of a group that's related to the absolute Galois group of $\mathbf{Q}_p$ (the Weil–Deligne group). This is called the Local Langlands Correspondence. The Global Langlands Correspondence is that a similar kind of relation holds between automorphic representations of $\mathrm{GL}(n,\mathbf{A})$ and $n$-dimensional representations of some group related to Galois group (the conjectural Langlands group). These correspondences should be nice in that things that happen on one side should correspond to things happening on the other. This fits into another part of the Langlands program which is the functoriality conjectures (really the correspondences are special cases). Basically, if you have two reductive groups $G$ and $H$ and a certain type of map from one to the other, then you should be able to transfer automorphic representations from one to the other. From this view point, the algebraic geometry side of the picture enters simply as the source for proving instances of the Langlands conjectures. Pretty much the only way to take an automorphic representation and prove that it has an associated Galois representation is to construct a geometric object whose cohomology has both an action of the Hecke algebra and the Galois group and decompose it into pieces and pick out the one you want.
As for suggestions on what to read, I found Gelbart's book Automorphic forms on adele groups pretty readable. This will get you through some of what I've written in the first two paragraphs for the group $\mathrm{GL}(2)$. The most comprehensive reference is the Corvallis proceedings available freely at ams.org. To get into the Langlands program there's the book an introduction to the Langlands program (google books) you could look at. It's really a vast subject and I didn't learn from any one or few sources. But hopefully what I've written has helped you out a bit. I think I need to go to bed now. G'night.
Best Answer
I'll be straying outside of my expertise here, so the following probably contains some errors. I'm just adding my perspective as someone who has (at one time at least) thought about some of this stuff. Hopefully this will help foster some discussion.
1) Modular curves are locally symmetric spaces, meaning they are of the form $\Gamma\backslash G/K$ where $G$ is (for simplicity) a semi-simple Lie group (e.g. $SL_2({\bf R}$), $K$ a maximal compact subgroup (e.g. $SO(2)$), and $\Gamma$ a discrete subgroup of finite index in $G({\bf Z})$ (e.g. $SL_2({\bf Z})$) (an "arithmetic" subgroup). Margulis' work on rigidity of lattices in Lie groups implies that unless $G$ is $SO(1,n)$ or $SU(1,n)$, $\Gamma$ is a congruence subgroup of $G({\bf Z})$ (a congruence subgroup is an arithmetic subgroup containing the kernel of the reduction map $G({\bf Z})\rightarrow G({\bf Z}/N{\bf Z})$). Note that $SO(1,2)\simeq SL_2({\bf R})$, $SO(1,3)\simeq SL_2({\bf C})$.
In my experience, people typically only talk about hyperbolic (or symmetric) spaces and fuchsian groups in passing because there is more information available in explicitly using $G$ and $\Gamma$ (So they may start out discussing e.g. a quotient of hyperbolic n-space by a discrete subgroup, but they prove things using that $H^n$ is really $SO(n,1)/SO(n)$ and $\Gamma$ is really a subgroup of $SO_{n,1}({\bf Z})$).
When $\Gamma$ is a congruence subgroup of $G({\bf Z})$, you can think of $\Gamma \backslash G/K$ as $G({\bf Q})\backslash G({\bf A})/K\cdot K_f$, where $K_f$ is a compact open subgroup of $G({\bf A_{\rm f}})$ This tends to be simpler to work with, since $G({\bf Q})$ has a simpler structure than $\Gamma$ (algebraic groups over fields instead of rings). Note that strange things can happen with noncongruence subgroups (e.g. there might not be any cusp forms).
Automorphic forms are certain functions on $\Gamma\backslash G$. Modular forms are classically defined on $G/K$, but you can do a little transform-and-lift to get them as certain "holomorphic" automorphic forms. With automorphic forms, the power of representation theory enters, and you have automorphic $L$-functions (and Hecke operators).
2) Modular curves are Shimura varieties, meaning (kind of) that $G/K$ has a complex structure, and so (after some work) $\Gamma \backslash G/K$ is an algebraic variety. More work shows that the Shimura variety is defined over a number field (the canonical model of the variety over the reflex field).
Thus you can attach to it a Hasse-Weil zeta function, which naturally factors as an alternating product of $L$-functions attached to the cohomology groups of the Shimura variety. These are supposed to be automorphic. In the modular case, the Eichler-Shimura relation makes this connection pretty simple to prove, but it is not known in general, and is really hard. (I'm not sure what the state of the art is. I'm pretty sure Hilbert modular varieties ($GL_2$ over a totally real field), Picard modular surfaces ($SU(2,1)$), and the next-simplest Siegel modular variety ($GSp_4$, thinking of $GL_2$ as $GSp_2$) are known. I think more general unitary groups are known, but I can't pin down exact statements).
The cohomology groups carry an action by $G({\bf A}_f)$, which gives rise to an action via Hecke operators (thinking of Hecke operators as members of the group algebra for $G({\bf A}_f)$). There are simpler ways to see this. The etale cohomology groups also carry an action by the absolute Galois group of the reflex field, which gives rise to $\ell$-adic Galois representations.
Modular/automorphic forms are sections of ("automorphic") vector bundles on $\Gamma \backslash G/K$. The algebraic structure on the Shimura variety has consequences for automorphic forms e.g. in terms of rationality of Fourier coefficients and special values of $L$-functions.