I wouldn't go too far with the entire "first quantization"/"second quantization" thing. You could imagine a free string field theory where the string isn't allowed to interact, but the nice thing about string perturbation theory is that the interactions and the propagation are different aspects of the same thing. This is in contrast to the perturbation theory of quantum field theories where the interactions and the propagators are different things (one is pointlike, the other is lifelike). The CFT (really, a theory of 2D quantum gravity) is what you start with, and you automatically get both the "free" string theory and the interactions. The question of "second quantization" (a term I really hate) is whether or not you can derive the formal power series resulting from adding the various amplitudes associated with Riemann surfaces of varying general as the perturbation expansion of another theory.
To answer your questions about how you go from fields on the worldsheet to fields in spacetime, you quantize the theory on the cylinder, and each vector in the Hilbert space corresponds to a spacetime field (because you can Fourier expand fields on the cylinder, this isn't as crazy as it sounds). However, because you really are doing 2D quantum gravity, you have to deal with the gauge invariances. The nice way to do this is using BRST quantization, and the actually physical fields are the cohomology of the BRST operator acting on the CFT Hilbert space.
This is pretty standard material in a first course on string theory. I don't have them on me, but I'd expect Eric D'Hoker's lectures in the IAS volumes on QFT and strings for mathematicians would do this.