Compressive sensing (CS) is a data acquisition and recovery technique for finding sparse solutions to linear inverse problems from sub-Nyquist measurements. CS features a wide range of computationally efficient and robust signal recovery methods, based on sparsity seeking optimization. In this paper, we present and analyze a class of sparse recovery algorithms, known as hard thresholding methods. We provide optimal strategies on how to set up these algorithms via basic ``ingredients'' for different configurations to achieve complexity vs. accuracy tradeoffs. Simulation results demonstrate notable performance improvements compared to state-of-the-art algorithms both in terms of data reconstruction and computational complexity.