Beam-target double-spin asymmetries and target single-spin asymmetries were measured for the exclusive pi(0) electroproduction reaction. gamma(*) p -> p pi(0), expanding an analysis of the. gamma(*) p -> n pi(+) reaction from the same experiment. The results were obtained from scattering of 6-GeV longitudinally polarized electrons off longitudinally polarized protons using the CEBAF Large Acceptance Spectrometer at Jefferson Laboratory. The kinematic ranges covered are 1.1 < W < 3 GeV and 1 < Q(2) < 6 GeV2. Results were obtained for about 5700 bins in W, Q(2), cos(theta(*)), and phi(*). The beam-target asymmetries were found to generally be greater than zero, with relatively modest phi(*) dependence. The target asymmetries exhibit very strong phi(*) dependence, with a change in sign occurring between results at low W and high W, in contrast to pi(+) electroproduction. Reasonable agreement is found with phenomenological fits to previous data for W < 1.6 GeV, but significant differences are seen at higherW. When combined with cross-sectional measurements, as well as pi(+) observables, the present results will provide powerful constraints on nucleon resonance amplitudes at moderate and large values of Q(2), for resonances with masses as high as 2.4 GeV.