In this work, the Friedmann equations, which represent the fundamental equations of cosmological models, are derived using a Newtonian and a relativistic approach by solving Einstein's field equations in a high level of detail. The space-time geometry in the form of the Friedmann-Robertson-Walker metric is derived and the calculations of the Christoffel symbols, the Ricci tensor and Ricci scalar, as well as the solution of the field equations are described in detail. The energy-momentum tensor assumes that matter in the universe behaves like an ideal fluid.The relationship between the different densities in the universe and the scale factor and the resulting three phases in the evolutionary history of the universe are explained. The time-varying ratio of matter density to vacuum density in the universe eventually led to the reversal of expansion, i.e., the change from a decelerated to an accelerated expansion of space. With the help of the second Friedmann equation and an equation for the expansion force, it is demonstrated at which density ratio and at what time this occurred. Assuming a flat universe and neglecting the radiation density, the Friedmann equation is solved and equations for the scale factor and the Hubble parameter are derived.Equations are derived to determine the cosmological horizons, the Hubble radius, and the worldlines of photons (light cones) and of stationary objects moving only within the Hubble flow. Using example calculations and their representations in space-time diagrams, the interrelations of these quantities are particularly elaborated.