Plethysm is an operation on symmetric functions, which is important in representation theory and algebraic combinatorics. However, efficient methods for finding the coefficients that emerge when plethysms of Schur functions are expanded using the Schur basis remain a challenge. The main goal of this thesis is to explore and suggest solutions to the problem of determining plethysm coefficients. This study aims to introduce methods for computing these plethysm coefficients, offering formulas where feasible and investigating their combinatorial and algebraic characteristics. This research seeks to employ theoretical and computational tools analysis to enhance our comprehension of plethysm and contribute to the wider domain of symmetric function theory.