Transport measurements on few-layer graphene (FLG) are important because they interpolate between the properties of single-layer graphene (SLG) as a true two-dimensional material and the three-dimensional bulk properties of graphite. In this article we present four-probe local charge transport and nonlocal spin-valve and spin-precession measurements on lateral spin field-effect transistors on FLG. We study systematically the charge- and spin-transport properties depending on the number of layers and the electrical back gating of the device. We explain the charge-transport measurements by taking the screening of scattering potentials into account and use the results to understand the spin data. The measured samples are between 3 and 20 layers thick, and we include in our analysis our earlier results of the measurements on SLG for comparison. In our room-temperature spin-transport measurements we manage to observe spin signals over distances up to 10 and measure enhanced spin-relaxation times with an increasing number of layers, reaching ps as a maximum, about times higher than in SLG. The increase of can result from the screening of scattering potentials due to additional intrinsic charge carriers in FLG. We calculate the density of states of FLG using a zone-folding scheme to determine the charge-diffusion coefficient from the square resistance . The resulting and the spin-diffusion coefficient show similar values and depend only weakly on the number of layers and gate-induced charge carriers. We discuss the implications of this on the identification of the spin-relaxation mechanism.