I recommend you to first have a good handle of Spinors in GR. I personally took this route in getting a grip on Spinors:

1. A Child's Guide to Spinors : http://www.weylmann.com/spinor.pdf

2. An Introduction to Spinors : https://arxiv.org/abs/1312.3824

3. Penrose, Roger (1960). "A spinor approach to general relativity"

The third one is going to be a smooth if you have gone through the first two documents which introduces to spinors in a wonderful way.

Then you can directly jump into these for further learning about Petrov Classification:

1. Petrov, A.Z. (2000). "Classification of spaces defined by gravitational fields". General Relativity and Gravitation. 32 (8): 1665–1685 (This is an English translation)

2. Hall, Graham (2004). Symmetries and Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics)