In this paper, we present a novel two-step inertial algorithm for finding a common fixed-point of a countable family of nonexpansive mappings. Under mild assumptions, we prove a weak convergence theorem for the method. We then demonstrate its versatility by applying it to convex minimization problems and extending it to data classification tasks, specifically through a multihidden-layer extreme learning machine (MELM). Numerical experiments show that our approach outperforms existing methods in both convergence speed and classification accuracy. These results highlight the potential of the proposed algorithm for broader applications in machine learning and optimization.