A lot of string-theory questions have been asked here. This one is, hopefully, different in that this inquiry is specific and the expected answer would be more mathematical than philosophical in character.
To give some context, I'm familiar with LQG, QFT etc. I'm a beginner in CFT and string-theory. On reading papers on AdS/CFT and String Theory the following sentence often appears: "The low-energy effective action for $x$ dimensional ST of type $y$ reduces to that of supergravity (SUGRA) in $z$ dimensions with matter fields of types $a$, $b$ and $c$".
The $64,000 dollar question any noob, such as me, wants answered is what are the mathematical/technical steps or arguments which lead us from the fundamental action of string-theory (the Polyakov/Nambu-Goto action) to a low-energy effective description. Of course, I might be asking a question for which the only concise answer is: "go read GSW". Here's hoping otherwise 😉
Best Answer
The basic answer is indeed "go read GSW or Polchinski" but if your goal was to get from the Polyakov action to a low-energy effective description as quickly as possible there are parts you could skip. In brief you want to
Work out the spectrum of the first quantized string. The "best" method is modern, covariant quantization, but the quickest method is light-cone quantization. If you want to know how to do this for say type IIA or IIB superstring you need to read GSW Chap. 2 and 4.3.
Deduce the existence of spacetime supersymmetry in the theory. This is the content of GSW Chap. 5.
There is a unique low-energy action with the massless field content and spacetime supersymmetry of type IIA or IIB string theory, namely the actions for IIA and IIB Sugra, so you know on general grounds that the action for IIA or IIB supergravity must be the low-energy effective action for IIA or IIB string theory.
If the general argument of 3. leaves you feeling uneasy you can remove this unease by computing a variety of 3 and 4 point tree amplitudes directly in string theory (now you need to read Chap. 7 of GSW), taking their low-energy limit, and comparing to the 3 and 4 point amplitudes you compute using IIA and IIB Sugra. You will find perfect agreement.
If you want to generalize this to include additional matter fields you will need to extend your analysis to the heterotic string and/or learn how to include D-branes into your type II analysis. Now you need to start reading Polchinski or various reviews, but you'll have a good place to start if you have done 1,2 and part of 4.