First of all some terminology. One usually talks about Podles spheres, since they are a one parameter family. If you say the Podles sphere you probably mean the one that is often referred to as the standard one. My indications will refer to the whole family.
You do not clarify your background and your directions. The family of Podles spheres has been used as an example in all possible meaning of the word quantization, I guess you could easily find a list of hundred of references about it. I'll stick to some paper that can be considered as foundational in various aspects.
First: as NC C*-algebra the first reading should be Podles original paper, which is simple and well written P.Podles Quantum spheres, Lett.Math.Phys. 14, 193-202 (1987).
From the point of view of quantum homogeneous spaces, i.e. -coideal subalgebras in Hopf--algebras I strongly suggest M.Dijkhuizen and T. Koornwinder Quantum homogeneous spaces, duality and quantum 2-spheres, Geom.Dedicata 52, 291-315 (1994).
In both these approaches, more or less evidently, q-special functions pop up at some point.
M. Noumi and Mimachi K., Quantum 2-spheres and big q-Jacobi polynomials, Comm. Math. Phys. 128, 521-531 (1990) is the one not to avoid.
Last but not least you may be interested in understanding Podles spheres as deformations of Poisson structures (say à la Rieffel), which is beautifully explained in A.Sheu, Quantization of the Poisson SU(2) and its Poisson homogeneous space - the 2 sphere, Comm. Math. Phys. 135, 217-232 (1991). I suggest that here you first read the appendix by Lu and Weinstein where the Poisson structures are explained neatly and simply, and then go the quantization part (some of which may result rather technical at first).
Then, of course, as mathphysicist mentioned, the whole issue of putting spectral triples opens up; the literature there is much more scattered (several attempts and several choices as well) and I guess one should just simply dive into open sea and see what happens...
ADDED:
Personally I would start with the paper by Dijkhuizen and Koornwinder that settles the algebraic part (generators and relations) and has a down-to-earth approach, without technicalities (if you know a little about Hopf algebras). I would not dismiss the paper by Noumi and Mimachi if you're interested in spectral triples. q-special functions means harmonic analysis on the sphere: it shouldn't surprise it is important if you look for Dirac operators satisfying some invariance condition.
Well, a lot of questions, some of which Theo already answered in a very nice way. Let me just give some additional remarks and hints how I think about DQ and Poisson geometry in relation to quantum physics.
Concerning the first question:
the good replacement (in view of Gel'fand duality) of a point on phase space is a (pure) state on the quantum algebra. While for $C^\ast$-algebras this is standard lore, in formal DQ things are slightly more tricky: of course you can argue that a formal star product yields not yet a quantum observable algebra as $\hbar$ does not have a "value" (say $1$ in your favorite unit system), so you should postpone the question of states till when you reach a "convergent/strict" DQ. This is often possible but in general completely unknown. Surprisingly, there is a good notion of states already for formal star products: essentially the same definition applies, take positive functionals of the algebra $C^\infty(M)[[\hbar]]$ which are $\mathbb{C}[[\hbar]]$-linear and take values in $\mathbb{C}[[\hbar]]$. To define positivity you make use of the fact that $\mathbb{R}[[\hbar]]$ is an ordered ring. Then many techniques of $C^*$-algebra theory can be carried over to this entirely algebraic framework. In fact, we have worked out many things like the GNS construction of representations etc.
Now the point is that a classically positive functional $\omega_0\colon C^\infty(M) \longrightarrow \mathbb{C}$ (which is a positive Borel measure with compact support by a smooth version of Riesz' Theorem) may no longer be positive with respect to a given star product $\star$. Thus you need to add higher order corrections $\omega = \omega_0 + \hbar\omega_1 + \cdots$ in order to gain positivity. It is a (quite non-trivial) theorem that this is always possible, even in a "differential" sense that all the higher orders are of the form $\omega_0 \circ D_r$ with some differential operator $D_r$.
You can apply this now to your favorite classical state, the delta-functional at a given point. The corresponding (non-unique) deformation is then the quantum analog of what a point can be, in some sense the best thing you can get.
The uncertainty principle can be understood as the reason why positivity fails for $\omega_0$ itself and why higher orders are necessary...
The second question: of course, for hard physical applications you only need $\mathbb{R}^{2n}$, maybe a cotangent bundle but that's it. Even a generic symplectic manifold is hard to motivate from this point of view.
But there are also reasons from physics why one should take care of DQ of more general Poisson manifolds:
a) Symmetries: whenever you have a classical symmetry encoded by a momentum map, then $\mathfrak{g}^\ast$ is a Poisson manifold. Quantizing a symmetry then amounts to quantize the momentum map in an appropriate way. There are several competing definitions but essentially all involve a DQ of the linear Poisson structure on $\mathfrak{g}^*$.
b) Aesthetics: to have a general framework in which you can discuss your relevant examples might be useful and open your view, even though the examples might be very very special inside this bigger framework.
c) Applications in NCG: many models of noncommutative space-times require more general Poisson structures to be quantized than just symplectic ones. It is even not clear that space-time allows for a symplectic structure at all, but it certainly carries interesting Poisson structures. In serious models of NC space-times, the Poisson structure itself should be treated as a dynamical quantity, i.e. a field. Then there is no reason why it should be non-degenerate everywhere. These models are of course all still very speculative...
d) Toy models: one can view complicated Poisson/symplectic manifolds as toy models for the infinite-dimensional phase spaces of classical field theories with gauge symmetries. Here the true phase spaces are sort of Marsden-Weinstein quotients (in ugly infinite dimensions) which can have quite generic geometry. So one tries to learn something about their quantization by looking at finite-dim models having at least also a complicated geometry.
Third question: Where is the Hilbert space...
After what I said for the first question, this is now pretty clear and follows the same line of argument as in AQFT: Having the algebra of observables, one takes a look at all $^*$-representations on, say pre-Hilbert spaces, by means of a GNS construction. The notion of pre Hilbert space works very much the same for ordered rings like $\mathbb{R}[[\hbar]]$. This has been worked out in detail in many places and gives indeed physically interesting results. The main advantage is now that one can take a look at different representations which can encode different physical situations...
OK, sorry for such a long blurp. I hope it gives some inspiration.
Best Answer
As far as I understand, the flag manifolds with Kahler structures mentioned in the question are simply coadjoint orbits of compact Lie groups with the Kirillov-Kostant-Souriau bracket, so their quantizations will yield quotients of the usual enveloping algebra $U(g)$ and will not have to do with quantum groups. I suppose that the q-spaces discussed in the question are meant to be q-deformations of these.
Here are some papers about this: arXiv:math/0206049 and arXiv:math/9807159. This is a rather subtle business: e.g., it is explained that the 2-parameter deformations (similar to the 2-parameter family of Podles spheres) do not always exist, although they do exist in type A and in many other cases, e.g. if orbits are symmetric spaces.