The initial value problems (IVPs) of Caputo fractional-order differential equations are very important in control systems modelling and simulation. A series of numerical algorithms are proposed in the paper in solving systematically various kinds of Caputo equations. For linear Caputo equations, the divergent problems of the existing Taylor auxiliary function are pointed out, and two effective algorithms are presented to transform the nonzero IVPs into zero ones, where the closed-form solutions are available. Furthermore, algorithms for nonlinear Caputo equation are also presented, aiming at finding numerical solutions to all kinds of Caputo equations. Error analysis for the proposed algorithms is provided, and numerical examples are presented to illustrate the accuracy and effectiveness of the algorithms.