We present the first data release of high-resolution ($\le 0.2$ arcsec) 1.5-GHz radio images of 103 nearby galaxies from the Palomar sample, observed with the eMERLIN array, as part of the LeMMINGs survey. This sample includes galaxies which are active (LINER and Seyfert) and quiescent (HII galaxies and Absorption line galaxies, ALG), which are reclassified based upon revised emission-line diagrams. We detect radio emission $\gtrsim$ 0.2 mJy for 47/103 galaxies (22/34 for LINERS, 4/4 for Seyferts, 16/51 for HII galaxies and 5/14 for ALGs) with radio sizes typically of $\lesssim$100pc.Weidentifytheradiocorepositionwithintheradiostructuresfor41sources.Halfofthesampleshowsjettedmorphologies.Theremaininghalfshowssingleradiocoresorcomplexmorphologies.LINERsshowradiostructuresmorecore-brightenedthanSeyferts.Radioluminositiesofthesamplerangefrom10$^{32}$ to 10$\mathrm{<mrow\; data-mjx-texclass="ORD">40</mrow>}$ erg s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$: LINERs and HII galaxies show the highest and the lowest radio powers respectively, while ALGs and Seyferts have intermediate luminosities. We find that radio core luminosities correlate with black hole (BH) mass down to $\sim$10$^{7}$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$, but a break emerges at lower masses. Using [O III] line luminosity as a proxy for the accretion luminosity, active nuclei and jetted HII galaxies follow an optical fundamental plane of BH activity, suggesting a common disc-jet relationship. In conclusion, LINER nuclei are the scaled-down version of FR I radio galaxies; Seyferts show less collimated jets; HII galaxies may host weak active BHs and/or nuclear star-forming cores; and recurrent BH activity may account for ALG properties.