I am just looking for a basic introduction to the Podles sphere and its topology. All I know is that it's a q-deformation of $S^2$.
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.
I am not an expert but perhaps the paper Spectral geometry: case of quantum spheres by Andrzej Sitarz could serve as a reasonably good starting point.