Start learning proof complexity
It depends on what kind of a "beginner" level you wish to have. I don't think there is a real good undergraduate level text on proof complexity (this is probably true for most specialized sub-areas in complexity). But for a beginner (graduate level) sources, I would recommend, something like understanding well the basic exponential size lower bound on resolution refutations of the pigeonhole principle (via random restrictions, width-size tradeoff and feasible interpolation), and to expand from that point further. This could be achieved (approximately) as follows:


Stasys Jukna, Extremal Combinatorics With Applications in Computer Science, 2001, Springer-Verlag, Section 4.8.
Eli Ben-sasson and Avi Wigderson, Short Proofs are Narrow - Resolution made Simple (2000), JACM.
P. Beame and T. Pitassi, Propositional proof complexity: Past, present, and future, Current trends in theoretical computer science: Entering the 21st century (G. Paul, G. Rozenberg, and A. Salomaa, editors), World Scientific Publishing, 2001, pp. 42--70.
Pavel Pudlák, Lower bounds for resolution and cutting planes proofs and monotone computations, The Journal of Symbolic Logic, vol. 62 (1997), no. 3, pp. 981-998. 


You can also consult the more self contained and lengthy text:


Peter Clote and Evangelos Kranakis, Boolean Functions and Computation Models (Chapter 5)


For the more logical side of proof complexity, as Kaveh suggested, you can start reading the first chapters of:


Stephen Cook and Phuong Nguyen, Logical Foundations of Proof Complexity (Perspectives in Logic, Cambridge Press, 2010).