Gas phase Elemental abundances in Molecular CloudS (GEMS) is an IRAM 30 m Large Program designed to provide estimates of the S, C, N, and O depletions and gas ionization degree, X(e-), in a selected set of star-forming filaments of Taurus, Perseus, and Orion. Our immediate goal is to build up a complete and large database of molecular abundances that can serve as an observational basis for estimating X(e-) and the C, O, N, and S depletions through chemical modeling. We observed and derived the abundances of 14 species (13CO, C18O, HCO+, H13CO+, HC18O+, HCN, H13CN, HNC, HCS+, CS, SO, 34SO, H2S, and OCS) in 244 positions, covering the AV ~3 to ~100 mag, n(H2) ~ a few 103 to 106 cm-3, and Tk ~10 to ~30 K ranges in these clouds, and avoiding protostars, HII regions, and bipolar outflows. A statistical analysis is carried out in order to identify general trends between different species and with physical parameters. Relations between molecules reveal strong linear correlations which define three different families of species: (1) 13CO and C18O isotopologs; (2) H13CO+, HC18O+, H13 CN, and HNC; and (3) the S-bearing molecules. The abundances of the CO isotopologs increase with the gas kinetic temperature until TK ~ 15 K. For higher temperatures, the abundance remains constant with a scatter of a factor of ~3. The abundances of H13 CO+, HC18 O+, H13 CN, and HNC are well correlated with each other, and all of them decrease with molecular hydrogen density, following the law ∝ n(H2)-0.8  ±  0.2. The abundances of S-bearing species also decrease with molecular hydrogen density at a rate of (S-bearing/H)gas ∝ n(H2)-0.6  ±  0.1. The abundances of molecules belonging to groups 2 and 3 do not present any clear trend with gas temperature. At scales of molecular clouds, the C18O abundance is the quantity that better correlates with the cloud mass. We discuss the utility of the 13CO/C18O, HCO+/H13CO+, and H13 CO+/H13CN abundance ratios as chemical diagnostics of star formation in external galaxies.