High signal to noise, high resolution spectra were obtained for a sample of normal, mild barium, and barium giants. Atmospheric parameters were determined from the FeI and FeII lines. Abundances for Na, Mg, Al, Si, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Sr, Y, Zr, Ba, La, Ce, Nd, Sm, Eu, and Gd, were determined from equivalent widths and model atmospheres in a differential analysis, with the red giant Eps Vir as the standard star. The different levels of s-process overabundances of barium and mild barium stars were earlier suggested to be related to the stellar metallicity. Contrary to this suggestion, we found in this work no evidence of barium and mild barium to having different range in metallicity. However, comparing the ratio of abundances of heavy to light s-process elements, we found some evidence that they do not share the same neutron exposure parameter. The exact mechanism controlling this difference is still not clear. As a by-product of this analysis we identify two normal red giants misclassified as mild barium stars. The relevance of this finding is discussed. Concerning the suggested nucleosynthetic effects possibly related to the s-process, for elements like Cu, Mn, V and Sc, we found no evidence for an anomalous behavior in any of the s-process enriched stars analyzed here. However, further work is still needed since a clear [Cu/Fe] vs. [Ba/H] anticorrelation exists for other s-process enriched objects.