I went through the first four chapters in a CFT notes by David Simmons-Duffin prepared for his lecture in Caltech, Ph229ab during the last few weeks.
What is Conformal Field Theory and Why do we study it?
In these four chapters, David explains nicely the motivation to study conformal field theory (CFT). We can start with very different microscopic theories, say, magnets system, liquid-vapor system or the so-called $\phi^4$ field theory, but end up with similar behavior at the critical point, where only the long wave-length physics matters. This is often known as universality.
This can be understood using an even simpler example, the central limit theorem in probability theory. It says when you add many independent random variables with the same distribution (the distribution itself can be arbitrary: uniform, Poison, Bernoulli, etc., whatever you like), the sum will end up to obey Gaussian distribution. That is the reason why we should learn Gaussian.
For the same reason, in order to understand phenomena like phase transition, instead of studying the various microscopic theory, we can study the underlying universal theory they all converge to. The underlying universal theory is often a CFT, where the physics looks exactly the same in different scales.
Path Integrals and Quantum-Classical Correspondence
In the second chapter, David does a good job explaining how a classical statistical system is related to a quantum system, and also how the path integral gives rise to difference quantization of the system if we choose different schemes. The discussion follows this QFT notes closely.
