In this paper, we derive closed-form exact and asymptotic expressions for the symbol error rate (SER) as well as channel capacity when communicating over the Fox's H-function fading channel. The SER expressions are obtained for numerous practically-employed modulation schemes in case of single as well as three multiple-branch diversity receivers: maximal ratio combining (MRC), equal gain combining (EGC), and selection combining (SC). The derived exact expressions are given in terms of the univariate and multivariate Fox H functions for which we provide a portable and efficient Python code. Since the Fox's H-function fading channel represents the most generalized fading model ever presented in the literature, the derived expressions subsume most of those previously presented for all the known simple and composite fading models. Moreover, easy-to-compute asymptotic expansions are provided so as to easily study the behavior of the SER and channel capacity at high values of the average signal-to-noise (SNR). The asymptotic expansions are also useful in comparing different modulation schemes and receiver diversity combiners. Numerical and simulation results are also provided to support the mathematical analysis and prove the validity of the obtained expressions.