Beam-target double spin asymmetries and target single-spin asymmetries in exclusive π+ and π− electroproduction were obtained from scattering of 1.6 to 5.7 GeV longitudinally polarized electrons from longitudinally polarized protons (for π+) and deuterons (for π−) using the CEBAF Large Acceptance Spectrometer (CLAS) at Jefferson Lab. The kinematic range covered is 1.1<W<2.6 GeV and 0.05<Q2<5 GeV2, with good anglular coverage in the forward hemisphere. The asymmetry results were divided into approximately 40,000 kinematic bins for π+ from free protons and 15,000 bins for π− production from bound nucleons in the deuteron. The present results are found to be in reasonable agreement with fits to previous world data for W<1.7 GeV and Q2<0.5 GeV2, with discrepancies increasing at higher values of Q2, especially for W>1.5 GeV. Very large target-spin asymmetries are observed for W>1.6 GeV. When combined with cross section measurements, the present results will provide powerful constraints on nucleon resonance amplitudes at moderate and large values of Q2, for resonances with masses as high as 2.3 GeV.