The spin structure functions g_1 for the proton and the deuteron have been measured over a wide kinematic range in x and Q2 using 1.6 and 5.7 GeV longitudinally polarized electrons incident upon polarized NH_3 and ND_3 targets at Jefferson Lab. Scattered electrons were detected in the CEBAF Large Acceptance Spectrometer, for 0.05 < Q^2 < 5 GeV^2 and W < 3 GeV. The first moments of g_1 for the proton and deuteron are presented -- both have a negative slope at low Q^2, as predicted by the extended Gerasimov-Drell-Hearn sum rule. The first result for the generalized forward spin polarizability of the proton gamma_0^p is also reported. This quantity shows strong Q^2 dependence at low Q^2, while Q^6\gamma_0^p seems to flatten out at the highest Q^2 accessed by our experiment. Although the first moments of g_1 are consistent with Chiral Perturbation Theory (ChPT) calculations up to approximately Q^2 = 0.06 GeV^2, a significant discrepancy is observed between the \gamma_0^p data and ChPT for gamma_0^p, even at the lowest Q2.